Skip to main content

Full text of "Relativity Theory Of Protons And Electrons"

See other formats 


DAMAGE BOOK 



^ 

w > ro 

m< OU_1 58925 5m 

^ CD X CO 



OSMANIA UNIVERSITY LIBRARY 

Call No. ^2,0 < ) Accession No, [^ ) 

Author 

Title 



4 
This book slwultf b* returned on or beforo the date 



^/ 
n or b 
last marked below, jp $? c 



RELATIVITY THEORY 

OF 
PEOTONS AND ELECTEONS 



LONDON 

Cambridge University Press 

FKTTJS& LANE 

NEW YORK TORONTO 
BOMBAY CALCUTTA MADRAS 

Macraillau 

TOKYO 

Maruzm Company Ltd 
All rights reserved 



KELATIVITY THEORY 

OF 

PKOTONS AND ELECTRONS 



by 
SIR ART II U R E I) I) I N (j TON 

M.A., D.Sc., LL.fiTRK.S:""" 

riuminn 1'rnJiMxtr al Jv/ ( <>in>inu <tnt{ Kriicnnn'Hlul 
I'lnlnioiilni in Ihc L'niwrsifti n( t'nnihrittu 1 ' 



CAMBRIDGE 

AT T1IE UNIVERSITY PRESS 
1036 



PRINTED IN GREAT BRITAIN 



PREFACE 

In this book I have endeavoured to give a connected account of a series of 
investigations in the borderland between relativity theory and quanjbi^H 
theory. It begins where my earlier book, The Mathematical Theory of 
Relativity, leaves off at the point where in our survey of nature we encounter 
the phenomenon of atomicity. To our gross senses matter seeing .continuous, 
and it has been treated as continuous in the usi^tl tftCory of relativity. 
Experiment has, however, taught us that it is composed of multitudes of 
units, and the theory is here extended to throw light on the existence and 
properties of these units. 

The central problem is to ascertain the conditions which fix the amount of 
mass and electric charge carried by protons and electrons. The present 
researches will probably be associated in the minds of many readers with 
the number 137; this is one of four numerical constants of nature for which 
the theory predicts definite values. Another fundamental constant, found 
to be 2.136.2 256 (approximately 3-150.10 79 ), can be described as the 
number of protons and electrons in the universe; but its practical importance 
is that its square root enters into the ratio of the electrical to the gravitational 
force between a proton and electron. The result of these determinations is 
that there are no arbitrary constants left in the scale of relations of natural 
phenomena. 

Besides giving concrete results of this kind, the theory has, I hope, thrown 
light on some of the obscure points in quantum theory and helped to deepen 
its foundations. I have sought a harmonisation, rather than a unification, 
of relativity and quantum theory. I do hot set out to obtain an all-embracing 
formula; but the investigation shows in detail how to combine the con- 
ceptions of the two theories in the solution of specific problems, which would 
be outside the range of either theory separately. 

The theory, as it was being developed, was published from time to time 
in the Proceedings of the Royal Society (121, p. 524; 122, p. 358; 126, p. 696; 
133, pp. 311, 606; 134, p. 524; 138, p. 17; 143, p. 327; 152, p. 253) and the 
Journal of the London Mathematical Society (7, p. 58; 8, p. 142) between 1928 
and 1935. But it has become increasingly difficult to deal with it in frag- 
ments. Much of it (including practically the whole of Chapters vi, xi and 
xvi) is now published for the first time. The new results of a practical kind 
include the theory of the Stern-Gerlach experiment, the theory of Bond's 
correction f|| to ejm, and the direct calculation of the number of particles 
in the universe. 



vi Preface 

In so extensive a work I cannot expect that serious mistakes have been 
entirely avoided. But now that the theory can be viewed as a whole, I think 
the reader will be convinced that there is a practicable way of progress along 
the lines I have attempted. I hope therefore that he will see in the im- 
perfections of this book an opportunity for developing, not an excuse for 
dismissing, the subject which it sets forth. 

Prof. G. Lomaitre and Prof. G. F. J. Temple have kindly read the book 
in proof. Their interest and criticism has encouraged me in the develop- 
ment of the theory, and I now owe them a further debt for many helpful 
suggestions. 

A. S. E. 

CAMBRIDGE, 
June 1936. 



CONTENTS 

INTRODUCTION l"*ge 1 

PART I. WAVE-TENSOR CALCULUS 

CHAPTER 1 Tensors and Matrices 13 

II The Sixteenfold Frame 20 

IK The Resolution of Matrices 34 

IV Space Vectors 60 

V Tho Simple Wave Equation 02 

VI Reality Conditions 75 

VII Strain Vectors and Phase Space 92 

Vlll The Differential Wave Equation 115 

IX The Hydrogen Atom 140 

X Double Wave Vectors 154 

PART II. PHYSICAL APPLICATIONS 

XI The Ricmann-ChristoffelTensor 179 

XII The Mass-ratio of the Proton and Electron 212 

XIII Standing Waves 229 

XIV The Cosmical Problem 256 
XV Electric Charge 281 

XVI The Exclusion Principle 308 

INDEX 331 



INTEODUCTION 

0*1. In 1928, P. A. M. Dirac made a bridge between quantum theory 
and relativity theory by his linear wave equation of the electron.f This 
is the starting point of the development of relativity theory treated in 
this book. 

Previously there had been three principal stages of progress, namely 
Einstein's special theory (1905), his general theory (1915), and Weyl's 
theory of relativity of gauge (1918). Summarising the state of the theory in 
1923, I wrote J 

We offer no explanation of the occurrence of electrons or of quanta; but in 
other respects the theory appears to cover fairly adequately the phenomena of 
physics. The excluded domain forms a large part of modern physics, but it is 
one in which ail explanation has apparently been baffled hitherto. The domain 
here surveyed covers a system of natural laws fairly complete in itself and 
detachable from the excluded phenomena, although at one point difficulties arise 
since it comes into close contact with the problem of the nature of the electron. 

Relativity theory was in fact as comprehensive and as logically complete 
as a purely macroscopic theory had any right to be. The next important step 
must be an extension to cover microscopic phenomena, or a unification with 
existing microscopic theories. 

Microscopic physics was the province of quantum theory; but in 1923 this 
was little more than a collection of empirical rules which led to no coherent 
outlook. The "new quantum theory" began with Heisenberg'n researches 
in 1925, and with the aid of many contributors it reached soon afterwards 
the current form generally called wave mechanics. The conditions were 
becoming ripe for a unification with macroscopic relativity theory. 

To say that Dirac's wave equation was the first connecting link gives only 
a partial idea of its importance. It was a challenge to those, who specialised in 
relativity theory. Dirac's object was to obtain a form of equation (fulfilling 
certain requirements of quantum theory) which should be invariant for 
rotations and Lorentz transformations. We had claimed to have in the 
tensor calculus an ideal tool for dealing with all forms of invariance and 
covariance. But instead of using the orthodox tool Dirac proceeded by a 
way of his own, and produced an expression of very unsymmetrical appear- 
ance, which he showed to be invariant for the transformations of special 
relativity theory. Why had this type of invariance eluded the ordinary 
tensor calculus? As C. G. Darwin put it, "it is rather disconcerting to find 

t Proc. Hoy. 8oc. A, 117, 610 (Feb. 1928). 
j Mathematical Theory of Relativity, p. 237. 



2 Introduction [<M 

that apparently something has slipped through the net ".f It was Darwin's 
insistence on this point in private conversation which led me to take up 
these investigations. 

The failure of ordinary tensor calculus to include Dirac's type of invariance 
is due to the introduction, at an early stage, of a convention whose arbi- 
trariness had already been noticed. J The analytical theory of tensers had 
been applied to physics by identifying its basic vector with a geometrical 
displacement (dx)P. By a change of application, namely by identifying the 
basic vector with Dirac's four-valued quantity 0, we obtain a new tensor 
calculus, here called wave-tensor calculus, in which the invariance of the 
wave equation falls into order. Formulae have to be found for expressing 
the old tensors (space tensors) in terms of the wave tensors; and this leads 
to a chain of new developments which have no counterpart in the tensor 
calculus of ordinary relativity theory. 

I was soon convinced that this was the extension of relativity theory for 
which we had been waiting, and that Dirac's equation was only the beginning 
of a more far-reaching application of the methods and conceptions of 
relativity theory to microscopic phenomena. After seven years' work I find 
the possibilities latent in the new departure still far from exhausted. 

Naturally others besides myself were attracted to the new opening. 
Allowing for divergences in the point of view, my first paper, dealing with 
formal developments including the rudiments of wave-tensor calculus, was 
perhaps not materially different from several other investigations published 
about the same time. || But a month or two later I camo across a clue to the 
origin of the charge of electrons and protons, ^f The trail has led all round the 
universe, so that the subject with which I began comes almost at the end of 
this book (Chapter xv). Ultimately the problem of the origin of charge was 
found to be inseparable from the problem of the origin of mass. I was thus 
led into a special field of investigation which, I think, has not been explored 
by other writers. 

Dirac's wave equation has led to important advances in quantum theory; 
but here we shall be working mainly on the relativity side of the bridge. It is, 
of course, impossible to treat protons and electrons without introducing a 
considerable amount of quantum theory. But its subordinate position will 
be apparent from the fact that the problems treated in Part II of this book 
are not touched upon in books on quantum theory; they depend essentially 
on developing the consequences of the relativistic conception. 

f Proc. Roy. Sac. A, 118, 664. This paper was of great assistance in my early work. 

j Mathematical Theory of Relativity, p. 49. Proc. Roy. Soc. A, 121, 524. 

|| I think that the most far-reaching, as well as the earliest, paper of this type was by 
J. v. Neumann, Zeits.fur Physik, 48, 868. But I have been more influenced by H. Tetrode, 
ibid. 50, 346, whose point of view was less unfamiliar to me. 

If "The Charge of an Electron", Proc. Roy. Soc. A, 122, 358 (Dec. 1928). 



0-2] Introduction 3 

0*2. As the work proceeded, it became focused on one problem, namely 
the origin of the four numerical ' c constants of nature ' ' . Seven fundamental 
constants are commonly recognised: 

m e the mass of an electron, 

m p the mass of a proton, 

e the charge of an electron, 

h Planck's constant, 

c the velocity of light, 

K the constant of gravitation, 

A the cosmical constant. 

Between these we must eliminate our arbitrary units of length, time and 
mass; we are then left with four purely numerical ratios. The most familiar 
are the mass-ratio m p /m e> and the fine-structure constant Ac/2?re 2 ; these are 
found in Chapters xir and xv. The value of *c, i.e. its ratio to a constant of 
similar dimensions furnished by the other constants, is obtained in Chapter 
xiv. Also, with the help of the other constants we replace A by a number N, 
the " number of particles in the universe", whose theoretical value is found 
in Chapter xvi. Thus all four constants are obtained by purely theoretical 

calculation.! 

The number of dimensions of space-time might be regarded as a fifth 
natural constant. Even this number is found to be determined unam- 
biguously by the epistemological principle that we can only observe 
relations between two entities ( 16-8). At a much earlier stage (Chapter vi) 
we prove that a four-dimensional neutral domain necessarily has the 
signature 3+1. 

So far as I can make out, the values of the constants given by this theory 
are in full agreement with observation. For three of the four constants the 
observations are accurate enough to provide a very stringent test. It would 
have been disconcerting if it had turned out otherwise; but the theory does 
not rest on these observational tests. It is even more purely epistemological 
than macroscopic relativity theory; and I think it contains no physical 
hypotheses certainly no new hypotheses to be tested. All that we require 
from observation is evidence of identification that the entities denoted by 
certain symbols in the mathematics are those which the experimental 
physicist recognises under the names "proton" and "electron". Being 
satisfied on this point, it should be possible to j udge whether the mathematical 
treatment and solutions are correct, without turning up the answer in the 
book of nature. My task is to show that our theoretical resources are 

t A general account of the principles on which the calculations are based is given in 
New Pathway* in Science, Chapter XL 

1-3 



4 Introduction [0-2 

sufficient and our methods powerful enough to calculate the constants 
exactly so that the observational test will be the same kind of perfunctory 
verification that we apply sometimes to theorems in geometry. 

The replacement of four empirical natural constants by calculated num- 
bers implies a unification of theory. Tri Maxwell's unification of electro- 
magnetism and optics the ratio of the electromagnetic to the electrostatic 
unit of charge was found to be equal to fche velocity of light ; similarly in the 
unification of macroscopic and microscopic theory the macroscopic con- 
stants K, A are found to be expressible in terms of m p ,m e ,e. The elimination 
of superfluous constants is an outward sign of the unification achieved; and 
for this reason I have regarded it as the first goal. But the theory that has 
taken shape in these investigations should supply a foundation for the 
treatment of other microscopic problems for which current quantum theory 
is insufficient. I have not pursued these further developments, partly because 
they often require a knowledge of the more technical side of quantum theory 
which I do not possess, and partly because the completion of the calculation 
of the four natural constants has seemed an appropriate stage at which to 
assemble the theory into a connected form. By way of exception, I have 
applied the theory to the Stern-Gerlach experiment (12-8); the result 
agrees with observation. 

0-3. The marriage of relativity theory and quantum theory should be a 
fruitful union as well as a formal union. In regard to the numerous formal 
unified theories that have been suggested, we may recall 

There are nine and sixty ways of constructing tribal lays, 
And every single one of them is right ! 

But T think they have been inspired by a fundamentally different conception 
of the problem of unification from that which I shall follow. There is a con- 
siderable amount of formal theory in this book; but it has been developed 
concurrently with the physical theory in Part II; and its progress has been 
guided as much by the definite applications, in which it was to be used, as 
by formal considerations. 

A unified theory does not necessarily mean a unified formula. The latter 
kind of unification is exemplified by the theory of the "Generalised Astro- 
nomical Instrument" which combines in a single equation the theory of the 
altazimuth, meridian circle, prime vertical instrument, equatorial and 
almucantar.f Such compression appeals more to the mathematician than to 
the physicist. We do not aim at producing a formula which shairinclude 
simultaneously the irregular gravitational fields of general relativity and 
the quantised energy of an atom. We seek instead the common meeting 
point from which the specialised developments and approximations appro- 
t ManMy Notices, H.A.S. 68, 171. 



0-4] Introduction 5 

priate to the gravitational and the atomic problem branch off. The source 
of quantum phenomena is a degeneracy, or exceptional integrability, which 
is associated with uniformity and symmetry, and invalidates the assumptions 
underlying the ordinary theory of macroscopic averages. It is therefore in 
uniform conditions (spherical space) that the linkage of quantum theory to 
macroscopic relativity theory must primarily be studied. 

For this reason we have not much to do with the formulae of general 
relativity, though we have much to do with its principles. We generally treat 
space-time of uniform curvature either the de Sitter form hyperbolic in 
the time dimension, or the Einstein form cylindrical in the time dimension, 
the space being spherical in either case. Thus, on the relativity side, we halt 
at a stage intermediate between the general theory and the special relativity 
theory of flat space. 

For the same reason, when gauge transformations are employed, the 
formulae used are those of Weyl's theory, not the author's generalisation 
of it.| The generalisation would be required if we dealt with space-time of 
irregular curvature; it coalesces with Weyl's theory in the uniform conditions 
here considered. 

With flat space-time we have nothing to do. The theory of space-time 
will here be developed pari passu with the theory of the material systems 
which occupy it. in this mode of approach the conception of infinite flat space 
never arises; it could not be brought into the theory except as a limit that 
might be approached but never attained. But in the same way the concep- 
tion of definitely empty space never arises; it could not be brought into the 
theory except as a limit that might be approached but never attained. 
(By definitely empty we mean that the probability of containing a particle or 
photon is zero.) Our rejection of flat space-time does not depend on the view 
that definitely empty space has a natural curvature determined by the 
cosmical constant.^ Space appears in our theory as the domain of the pro- 
bability distribution of a particle, so that it is an essential characteristic 
of space that it is occupied or has a finite a priori probability of being 
occupied; and it is non-controversial that it will have a curvature (or an 
expectation- value of the curvature) corresponding to the energy tensor (or 
expectation-value of the energy tensor) of its contents. 

0'4. I think it will be found that the theory is purely deductive, being 
based on epistemological principles and not on physical hypotheses. But 
it could not be presented in purely deductive form which would mean, 
I suppose, that it was treated as an investigation in pure mathematics with 
a physical denouement in the last chapter. It has seemed essential to 

t Mathematical Theory of Relativity, Chapter vu, Pt. II. 
j We are led to reject this view ( 11*7). 



6 Introduction [0-4 

keep the physical applications in mind throughout; for this purpose 
results must sometimes be anticipated which are not reached by deduc- 
tion until much later in the book, and interpretations must be employed 
which are not definitely established until the whole theory is connected 
together. 

This method gives rise to certain difficulties. For mathematical seasons 
we have to begin with the simplest equations; but these correspond to 
highly idealised systems to which the ordinary physical conceptions only 
partially apply. It is indeed obvious that a system must attain a consider- 
able degree of complexity before anything remotely resembling the ordinary 
method of observation is applicable to it. Consequently the physical ideas 
can only take shape gradually as we proceed. Space-time of a kind first 
appears in Chapter iv; but it has the wrong signature, and its scale is much 
too small. These defects become rectified as the developments in later 
chapters take us closer to actuality. So with most of the physical concep- 
tions; we .have to introduce preliminary notions before the theory is suffi- 
ciently advanced for a full treatment. The reader will probably find that 
many of the difficulties that occur to him in reading the earlier chapters 
arise from the unnatural conditions postulated in the most elementary 
equations, and that they resolve themselves automatically when the theory 
reaches more realistic problems. 

Those who are expert in quantum theory should bear in mind that we are 
proceeding from another starting point from that usually adopted. It 
would be foreign to my intention of developing the theory as a pure deduction 
from relativistic principles to transfer conclusions, however widely accepted, 
from the usual quantum theory which contains a large empirical element. 
Nevertheless, I make frequent appeals to current quantum theory for three 
purposes. Firstly, because it contains the definitions of the quantities with 
which I am concerned. It would be impossible to make a theoretical deter- 
mination of the constant known as the "mass of an electron" without an 
examination of the equations by which the quantum physicist has chosen 
to define it. Secondly, where the present theory coalesces with current 
theory, it is unnecessary to repeat purely analytical investigations which 
equally apply to either theory. Thirdly, certain results (especially the 
Exclusion Principle), which cannot be treated until late in the book, have 
been borrowed from current theory in anticipation. 

It may be well to make it clear that although the present theory owes 
much to Dirac's theory of the electron, to the general coordination of quan- 
tum theory achieved in his book Quantum Mechanics, and to the many 
contributions of himself and others on these lines, it is not " Dirac's Theory " ; 
and indeed it differs fundamentally on most points which concern relativity. 
It is definitely opposed to what has commonly been called "relativistic 



0-5] Introduction 7 

quantum theory ", which, I think, is largely based on a false conception of 
the principles of relativity theory. 

Atomic nuclei and free neutrons are outside the scope of this book. I see 
no reason to fear that they will not fall into place in the theory; but I have 
not developed any ideas on this point far enough to be worth recording. In 
the main the theory of radiation has also been excluded; but there are three 
short references ( 9-5, 14-3, 16-3) which show how it might be approached 
in the present treatment. 

0* 5. The division of the book into two parts, the one ostensibly treating 
the auxiliary mathematical calculus and the other the physical applications, 
is only a rough separation. Physical interpretations are considered as early 
as possible; but with the introduction of double wave tensors in Chapter x 
the relation of the mathematics to the physics changes considerably. 
Instead of starting with the mathematical result and interpreting it as far 
as possible physically, we start with the physical problem and formulate it 
mathematically. The auxiliary mathematical development still continues; 
but it is now guided by the character of the physical problems for which its 
aid is required. 

The student of relativity theory may well feel a grievance at the turn 
which the auxiliary mathematics has taken. The macroscopic theory seemed 
to indicate that Differential Geometry was the key to world-structure. After 
being at pains to acquire some familiarity with this subject, we find that all 
the new advances depend upon modern Algebra. The algebra required in 
the present book is developed practically ab initio by old-fashioned methods 
which, I fear, betray my limitations as an algebraist/though they may make 
the theory more accessible to those most interested. Let me freely admit that 
ability to use the more powerful modern algebraic methods would be an im- 
mense advantage in handling these problems. For the kind of algebra chiefly 
required I have found most helpful C. C. MacDuffee, The Theory of Matrices J[ 

A few remarks on terminology, etc. may be useful. I would direct special 
attention to the limited use of the summation convention (p. 22), to my 
unorthodox use of the term "algebraic" (p. 20), and to the change in the 
order of writing the suffixes of the Riemann-Christoffel tensor (p. 181). 
I would emphasise that wave analysis is a method, not a theory, and may be 
applied to any physical tensor; therefore statements about the physical 
meaning of the various products of wave analysis necessarily refer to some 
special application (singled out by custom) and are not of general validity. 

The term universe is used so often as perhaps to suggest megalomania. 
It is really the opposite of megalomania, for it takes the place of infinity in 
elementary wave mechanics. Mathematically it is much easier to treat a 

t Ergebnisse der Mathematik und ihrer Grenzgebiete (Julius Springer, 1933), 



8 Introduction L' 5 

whole universe than part of one the universe being, of course, idealised to 
accord with the simple conditions postulated in elementary problems. It is 
more elementary to suppose that the uniform conditions continue inde- 
finitely than to terminate them by a physical barrier; supernatural barriers 
are often misleading, and should be avoided if possible. In the earlier 
chapters there is sometimes a difficulty in deciding whether our equations 
refer to an electron lor to the universe. But the fact is that the electrons treated 
in ordinary elementary quantum theory are very much like the universe 
only bigger. They are said to be "infinite plane waves". No doubt it is 
intended that they shall be replaced by waves of more reasonable dimensions 
in practical approximations; but this applies also to our theory. For applica- 
tions in which a millimetre is a good enough approximation to infinity, it is 
a fortiori a good enough approximation to 400 megaparsccs. 

By a particle, I mean, not a classical particle, but a conceptual entity 
whose probability distribution is specified by a \vave function. At different 
stages in this book, different applications of wave analysis are made; and the 
corresponding particles have different properties. In the earlier chapters 
the particles are rudimentary protons and electrons existing in the rudi- 
mentary space-time there treated. They gradually develop into recognisable 
electrons and protons in macroscopic space-time, when the theory is ex- 
tended far enough to introduce the observable relations by which protons 
and electrons are known experimentally. The reader should therefore not 
be surprised to find that initially the positive and negative particles have 
completely symmetrical properties; that is merely another illustration of 
the fact that the most elementary equations imply highly idealised con- 
ditions to which the ordinary conceptions of physics only partially apply. 

0*6. Those who have followed the progressive development of the 
theory during the last eight years may desire a comparison of the present 
revised theory with earlier versions. The papers are numbered for reference 
as folio ws:f 

1 . "A Symmetrical Treatment of the Wave Equation " , R.S. 121 , 524, 1 928 
Tl. "The Charge of an Electron", R.S. 122, 358, 1928. 

III. "The Interaction of Electric Charges", R.S. 126, 696, 1930. 

IV. "The Properties of Wave Tensors", R.S. 133, 311, 1931. 

V. "The Value of the Cosmioal Constant", R.S. 133, 605, 1931. 
VI. "The Mass of a Proton", R.S. 134, 524, 1931. 
VTI. "Sets of Anticommuting Matrices", M.S. 7, 58, 1931. 
VIII. "Theory of Electric Charge", R.S. 138, 17, 1932. 
IX. "The Factorisation of ^-Numbers ", M.S. 8, 142, 1933. 
X. "The Masses of the Proton and Electron", R.S. 143, 327, 1933. 
XI. "The Pressure of a Degenerate Gas, and Related Problems ", R.S. 152, 
253, 1935. 
t M.S. refers to Proc. Roy. Soc. and M.S. to Journ. Land. Math. Soc. 



0-6] Introduction 9 

I will begin with a definite withdrawal. In I it was suggested that the 
adjoint tetrads A\, E 2 , E z , E and JS? 15 , JK 25 , jB 35 , # 45 correspond to electrons 
of opposite spin, and this niisjudgment persisted in 111. The present view 
( 4-3, 4-4) of the role of E 5 was first reached in IV. I think there is no 
other point on which I went so completely astray; the other lines of 
development begun in this series of papers, though sometimes requiring 
substantial amendment, contain advances which in principle have been 
retained. The papers fall into three groups: 

(a) Auxiliary Mathematics (I, IV, VIT, IX). The ^'-symbols were at 
first defined so that A^ 2 = 1 ; the present notation, with E^ 1, begins 
in IV. In IV the mathematical development remains satisfactory; but 
the physical interpretation was confused, because a degenerate wave 
tensor was used in a context where the later results substitute a non- 
degenerate wave tensor. The proofs of two important theorems in VII 
and IX, namely the composition of a pentad by three imaginary and two 
real matrices, and the standard forms of pure wave tensors, have been 
found to be imperfect. Amended proofs are given in 3-5, 5-5. 

(b) Electric Charge (11, III, V11I). Paper II has been affected less than 
most of the early papers by subsequent progress, and can be regarded HH 
substantially correct so far as it goes. Gauiit's form of the matrix co- 
efficient of the Coulomb energy, employed in the paper, is now obsolete; 
but this scarcely affects the investigation at the stage concerned. The 
factor 136 was changed to 137 in subsequent papers; but the difference 
is a question of definition ( 15-9). Paper 111 represents an interim stage 
in a complicated investigation, and has the defects of an interim report. 
Progress remained unsatisfactory until the interchange energy was associatec 1 
with the operator P instead of with E$. The theory in Vlli is substantially 
the same as the second of the two methods given in this book ( 15-7). 

(c) Origin of Mass (V, VI, X, XI). With regard to V and VI, which 
were preliminary papers, 1 need only say that the present theory follows 
the ideas there suggested. In X the argument now replaced by the 
formula jR 2 = JS./2' was unsatisfactory; otherwise the changes are mainly 
of the nature of amplification. Paper XI is practically up to date ; but 
a numerical change was made necessary by the discovery of an in- 
consistency of a factor 2 in current quantum theory ( 9-6). 

The various lines of investigation were very much interlocked ; a back- 
ward state of one prevented progress in the others. Thus the whole work 
reached completion as one unit. The cosmical problem treated in XI was 
the last item on the main programme; and, after it was solved, there was 
not much difficulty in supplying the remaining investigations needed to 
fit together all the material. The investigations in the published papers 



10 Introduction [0-6 

are, of course, considerably altered in form now that they are connected 
to a homogeneous theory instead of to fragments of current theory 
partially modified to suit the growth of ideas. 

It was found in III that the theory offers an explanation why one 
dimension of the world differs from the other three; but, except for this, 
little attention was paid to reality conditions in the series of papers. 
This was deliberate; because it seemed premature to try to formulate 
reality conditions before the main lines of connection of the analytical 
theory with observational phenomena were settled. It was not until after 
the last of the published papers that I took up the problem and reached 
the reality conditions formulated, and used extensively, in this book. 

Strain vectors first appeared in X, but were to some extent anticipated 
in VIII. The more extended use of strain vectors, and the systematic 
discrimination between internal and external wave functions, is a feature 
of the new treatment. Other portions of the theory scarcely touched on 
in the published papers are 8-4-9-1, 10-4-11-9, 12-6-12-8, 15-8-16-9. 

At one time 1 laid stress on a suggestion (due to Zanstra) that the 
packing ratio in helium is an approximation to f|. It seemed likely that 
the binding of particles in a rigid nucleus might be represented as the 
loss of one degree of freedom of the double wave functions, with a 
corresponding reduction of the energy required for statistical equilibrium. 
Recent atomic weights make the packing ratio less close to J-|| than was 
at one time supposed; but in any case the theory of a rigid nucleus was 
not expected to apply to helium exactly. I still regard the suggestion as 
plausible; but as my investigations have not dealt with nuclear structure, 
the question remains in suspense. 



PAET I 
WAVE-TENSOR CALCULUS 



CHAPTER I 
TENSORS AND MATRICES 

1*1. Linear Transformations . 

A physical system may be described in many alternative ways. Different 
systems of coordinates may be used for specifying its position; different 
systems of units may be used for the measurement of mass, length, time; 
and so on. Accordingly our attention is directed to the problem of comparing 
systems of description in which there is a one-to-one correspondence between 
quantities A, B, C, ... occurring in one description and quantities .4', B 1 ', 
C', ... occurring in another description. 

The description commonly includes sets of associated quantities which 
are regarded as "components" of a single entity, e.g. the three components 
of a force. We then have a correspondence between an array of n quantities 
Ap in one description and A^ in another description (/u = 1, 2, ... n). 

We proceed at once to a special case of great importance, viz. when A ' is 
given by a linear transformation of A^ 

...+q lH A n [ 

...+?i,i4J 

etc. Using the summation convention of the tensor calculus, these formulae 
are written more compactly A > A /i i\ 

r A a =9oii A n (M2) 

and the transformation, or change of description, is described as A -> g A . 

The array of coefficients q afJL defines the change of the system of descrip- 
tion, so far as the characteristic A^ is concerned. Linear transformations 
possess the Group property ; that is to say, the resultant of a succession of 
linear transformations is a linear transformation. Thus we can have a set 
of systems of description such that, in passing from any one description to 
any other, the transformation of A^ is always linear. When for all systems 
of description contemplated the transformation of A^ is linear, A is called 
a tensor. 

By solving equations (Ml) we can find A l9 A 2 , ... in terms of -4/, A 2 ' , .... 
The resulting formulae are linear and may be written 

4r = ?o/-V- (M3) 

The array of coefficients q a ^ defines the in verse transformation to that defined 

fcysv 

If Bp is another array of n quantities occurring in the description of the 
physical system, and in the change of description in which A Q ->q aiL A tli 



14 Wave-tensor Cakulus [M 

Bp is said to be a tensor of the same kind as A^ (or to be cogredient with 

4>- 

If Cp is another array of n quantities occurring in the description, and in 
the change of description in which A a ->q afJt A^ 

C.+qr'Cp, (M5) 

C^ is said to be a tensor of opposite kind to A p (or to be contragredient to A^) 9 
Note the inversion of the order of the suffixes of q'. 

From Ap and C^ we may form an array of n 2 quantities A^G V which 
follows the transformation law 

^C^q^A^C^q^q^A^. (M6) 

If T^ v is an array of n* quantities occurring in the description, and in the 
change of description in which A^-^q^A^ 

T^q^'T^ (M7) 

(i.e. if it is transformed in the same way as A^C^ then T^ v is said to be a 
mixed tensor of the second rank of the class A^ . 

Tensor properties do not necessarily depend on the physical nature of 
the entity that is being described; they depend on the variety of descriptions 
which we admit. For example, the statement that B^ is a tensor of the same 
kind as A^ announces a limitation of the variety of description contemplated ; 
for there can be no compulsion to change our description of one physical 
feature of the system when the description of another feature is changed. 
But unless there is some systematic plan underlying our descriptions it will 
be impossible to assert any general laws governing the quantities occurring 
in the descriptions. 

For example, the strength of the wind is sometimes described by a 
number of dynes per square centimetre and sometimes by a number on the 
Beaufort scale. We cannot expect to find exact equations (relating our 
measures of the strength of the wind to other meteorological characteristics) 
applicable to both codes of measurement. By taking the wind strength to be 
a tensor of the class of tensors used for describing other meteorological 
characteristics we rule out one or other description not as illegitimate, but 
as unsuited to the purpose we have in mind, viz. to express the regularities 
underlying natural phenomena by mathematical equations governing the 
quantities which occur in our descriptions of the phenomena. 

1-2. Space Tensors and Wave Tensors. 

When the change of system of description includes a change of coordinates 
from (^ , a? 2 , # 3 , # 4 ) to (#/ , # 2 ' #3' > #4')* &* 1 infinitesimal coordinate difference 
dXp is transformed according to the formula 

dxi (1-21) 

* 



1-3] Tensors and Matrices 15 

etc. This may be written in the form (1-15) 

dx^q^'dXp (q^'-dx.'/dxj. (1-22) 

Thus every change of description contemplated as admissible corresponds 
to a linear transformation of dx^. Accordingly dx^ is a tensor; we call it 
a displacement vector. 

This is the basic tensor of the class of tensors used in the ordinary tensor 
calculus. Displacement vectors and all tensors of the same kind are called 
contravariant vectors; tensors of opposite kind are called covariant vectors. 
Mixed tensors of the same class are defined as in (M7); and more generally 
tensors of higher rank with 16, 64, 256, ... components are introduced, their 
transformation laws being 



We shall call this class of tensors space tensors. 

Thus, although the theory of tensors belongs primarily to the algebraic 
theory of transformations, it has usually been linked to geometry by 
identifying the basic tensor of the algebraic scheme with a geometrical 
displacement or coordinate difference dx^ . We shall here discard this special 
linkage. We shall introduce another class of tensors called wave tensors, 
derived from a basic contravariant wave vector ^ in the same way that the 
space tensors are derived from the basic contravariant space vector dx^ . 

For the moment we leave the basic wave vector unidentified. But at a 
certain point in the development of the system of wave tensors, we shall be 
able to side-step into a new class of tensors. On examining the properties 
of the new tensors we shall find that they can be identified with space 
tensors. Thus the wave-tensor calculus leads up to the ordinary space-tensor 
calculus and includes it as a side branch; but its greater comprehensiveness 
fits it to deal with certain entities in modern quantum theory which are not 
describable by space tensors. 

The basic wave vector will be identified in Chapter v. It turns out to be 
the four-valued wave symbol introduced into physics by P. A. M. Dirac in 
his linear wave equation of the electron. Vectors of this class cannot be 
reached from the ordinary calculus 6f space tensors, which does not begin 
far enough back. Our plan accordingly is to begin with these vectors, and 
lead up to the ordinary space vectors at a later stage. 

1*3. Chain Multiplication. 

Let Aj, Bj be two mixed tensors of the second rank. Having regard to the 
summation convention we recognise four different products 

A*B* 9 AJB V \ A*Bf % AjBf. (1-31) 

The first is the outer product, and the fourth is the inner or scalar product. 



16 Wave-tensor Calculus [1-3 

The second and third are called matrix prodticts and are denoted by AB and 
BA respectively. 

Matrix products are formed by chain multiplication, i.e. the second suffix 
of one factor is repeated as the first suffix of the succeeding factor (the 
repetition introducing a summation in accordance with the summation con- 
vention). The product AjB* is of this form. AjBf- is not a chain product 
as it stands; but it becomes one if it is rewritten as BfAj . 

On the understanding that chain multiplication is the only kind of 
multiplication admitted, no suffixes need appear in the formulae, since the 
reader can always supply appropriate suffixes when required. Thus the 
product of a number of double-suffixed quantities is written 

P = ABCD, (1-321) 

which stands for P^ABjCD. (1-322) 



This rule of multiplication is the distinctive feature of the matrix calculus. 
The notation is so useful that wo cannot afford to do without it. Nevertheless 
matrix calculus suffers from being more limited than tensor calculus; and 
we often want to introduce outer and scalar products and other combina- 
tions for which matrix calculus provides no notation. This necessitates 
resorting to various awkward shifts, and occasionally reverting to the full 
suffixed expressions. 

Chain multiplication does not contemplate quantities with more than two 
suffixes. We shall at first limit the term Sk matrix " to two-suffixed quantities 
representing two-dimensional arrays. Technically one-dimensional arrays 
are also matrices, but it would probably be confusing to include them. One- 
dimensional arrays will here be called vectors, even when no question of 
transformation properties arises. The term implies very little restriction so 
long as we do not specify the kind of vector. 

Chain multiplication cannot be carried beyond a vector, so that vectors 
can only occur at the beginning or end of a matrix product. We shall dis- 
tinguish initial vectors by an asterisk, final vectors being unmarked. This 
notation allows us to reintroduce outer multiplication to a limited extent. 
The rule is that, if it is impossible to interpret two symbols in juxtaposition 
as a chain product, they are to be interpreted as an outer product. Thus if 
0^ is a vector, the expressions 



are interpreted as (A$) xB, Ax 

where the symbol x indicates outer multiplication, chain multiplication being 

impossible after a final vector or before an initial vector. Or, with suffixes, 



1-4] Tensors and Matrices 17 

In particular, we have the following notation which is of great importance 

0X* denotes the outer product ^uX 



t \ ***t5 J 

denotes the scalar product x^u-J 

The asterisk is a substitute for suffix indications, and is dropped when the 
suffixes are inserted. 

A feature of matrix multiplication is that it is non-commutative; that is to 

sa y BA*AB. (1-34) 

It is to be remembered that the non-commutation only arises through the 
omission of suffixes; when suffixes are inserted in BA, the factors commute 

as usual. Thus iy*4/-4/V- ( 1>35 ) 

Since the suffixes are often omitted, we can no longer depend on dis- 
criminating contravariant from covariant vectors by the upper and lower 
positions of the suffixes. There would be little advantage in retaining a 
method of discrimination which only worked spasmodically. Accordingly, 
we shall in future generally write all wave tensor suffixes in the lower 
position. 

1*4. Transformation Laws of Wave Tensors. 

In 1-1 we introduced three kinds of tensors of the class A^ with trans- 
formation laws (1-14), (M5) and (1*17) respectively. The formulae may be 
written as B' aB C' = Ca ' T '= ' 

^a qa, J '> ^a u ixor > - L ar 



The products, as here written, are all chain products, so that the suffixes 
may be omitted and we have 

B' = qB, C*' = C*q', T' = qTq'. (1-41) 

Further by (1-12) and (MS) 



Therefore AJ = q^q^'A^ . (1-421) 

But A ' = *^A r ' 9 (1-422) 

where 8^ is the substitution operator, viz. 



= 0, 

Since A r f is an arbitrary array of four numbers, it follows from (1-421) and 
(1-422) that g^/^. (1-44) 

The left-hand side is a chain product; we can therefore drop the suffixes, 
obtaining ?? , = s (1 . 45) 



18 Wave-tensor Calculus [1-4 

In matrix calculus 8 has the algebraic properties of the number 1. For, 
if 8 is any matrix 88=8 88=8 

so that, dropping suffixes, 8/9 = $, 88 = 8. 

Accordingly 8 is called the unit matrix] and since it is equivalent to the 
number 1 in matrix calculus, we shall often denote it by 1, or with suffixes 
(1) . Then (1-45) becomes qq' = 1. Thus q' may be called the reciprocal of q, 
and it will sometimes be written as q- 1 . 

The formulae (1-41) and (1-45) constitute the principal transformation 
formulae in wave-tensor calculus. Summarising our results for reference, 
and changing to the notation which we shall usually employ, we have the 
following classification and nomenclature: 

Covariant (final) wave vectors 

i/j' = qtfj. (1-461) 

Contravariant (initial) wave vectors 



Mixed wave tensors T' = q Tq', (1-463) 

with ff?' = l- (1-464) 

These would reduce to the transformation laws of the ordinary tensor 
calculus if we set ^ = a.yaV, ^' = ^7^. <1') 

But, as already explained, the wave tensors are not linked to geometry in 
this way, and (1*47) does not apply. For a transformation of wave tensors, 
any matrix which has a reciprocal maybe used as q; that is to say, the corre- 
sponding transformation will give a new description which is included in 
the whole group of descriptions contemplated. 

A matrix which has no reciprocal is said to be singular. A singular matrix 
may be regarded as a generalisation of the algebraic number in much the 
same way that the unit matrix is a generalisation of the number 1 ; but there 
are infinitely many different singular matrices. As q approaches a singular 
value, one or more elements of its reciprocal q' tend to infinity; singular 
matrices q are therefore excluded in the foregoing transformation theory. 

1-5. Initial and Final Wave Vectors. 

The terms "initial" and "final", applied to wave vectors, define their 
behaviour in regard to chain multiplication, and do not necessarily describe 
their actual position in the sequence of factors (cf. (1-33)). As far as 
Chapter vi (inclusive) the initial vectors will be contravariant and the final 
vectors covariant. But it must not be supposed that this is a general rule, or 
that the asterisk is a symbol for contravariance. 



1-5] Tensors and Matrices 19 

In order to express the covariant transformation law (1*461) in a form 
appropriate for an initial covariant vector, we introduce a matrix q which is 
the transpose of 0, obtained by interchanging rows and columns; thus 

Then (1-461) stands for 

$* 

Hence, dropping suffixes, 0*' = 0*</. Treating (1-462) similarly, we have the 
transformation laws: 

Initial covariant wave vectors 

t*' = *li*q, (1-521) 

Final contravariant wave vectors 

x ' = q' x . (1-522) 

The outer product ift<f>* of two covariant wave vectors 0, $ is a covariant 
wave tensor S. Using (1-461) and (1-521) we obtain the transformation law: 

Co variant wave tensors S' = qSq. ( 1 53 ) 

These formulae will not be required until Chapter vn. 



CHAPTER II 
THE SIXTEENFOLD FRAME 

2* 1 . Symbolic Calculus . 

For our physical applications the significance of a matrix is embodied, not 
so much in its representation as an array of numbers, as in its non-commu- 
tative multiplication property (1*34). Most, if not all, of the properties of 
matrices which make them suitable for describing the conditions and 
activities of the physical universe are also possessed by general symbols 
endowed with the same non-commutative properties. 

We shall therefore develop a calculus containing a number of symbols 
which do not obey the commutative law of multiplication, but obey the 
other elementary laws of algebra. The following definitions are adopted: 

A symbol which commutes with every symbol in the calculus will be 
called etn algebraic number. 

The number 1 is defined to be a symbol which satisfies 

l.E = E.l = E, 

where E is any symbol in the calculus. From the definition of 1 the defini- 
tions of other algebraic numbers follow in the usual way. In particular i is 
defined to be a symbol satisfying 

iE = Ei, UE=-E. 

The underlying idea is that a symbol has no properties except such as 
are manifested by it in the operations of the calculus in connection with 
which it is used. Its nature lies in its behaviour; it has no intrinsic nature. 
Therefore if a symbol behaves like the number 1 in every possible operation 
of the calculus, it is the number 1. If our calculus is afterwards extended by 
the introduction of additional symbols or operations which give a further 
opportunity for discriminating behaviour, some of the symbols originally 
counted as algebraic may cease to be algebraic. We may regard "algebraic " 
as a relative characteristic depending on the range of symbols which con- 
stitutes our calculus. 

I have here deviated from the terminology in pure mathematics, where it 
is customary to give a much wider meaning to the term "algebraic". But 
I think that most readers of a physical treatise will naturally understand 
"algebra" to mean "ordinary algebra"; and therefore the distinction 
between quantities which obey the rules of ordinary algebra (including the 
commutative law of multiplication) and those which do not is most in- 
telligibly described by the adjectives "algebraic" and "non-algebraic". 



2-2] The Sixteen/old Frame 21 

2*2. Complete Orthogonal Sets. 
Let E l9 E 2 , jE 3 , JP 4 be four symbols which satisfy 

jy=-l, E^-E^ (^=1,2,3,4;^,,). (2-21) 

That is to say, the symbols are four mutually anticommuting square roots 
of 1. We shall find in 3*2 that there exist matrices which satisfy (2-21), 
so that we need have no qualms as to the legitimacy of postulating such 
symbols. 

When we are given an even number of anticommuting square roots of - 1 , 
we can always find an additional anticommuting square root, making the 
total number odd. Let iE^E^E^. (2-22) 

We have (iE 5 )* = E^ E 2 E 3 E^ E 2 E 3 ff 4 

_ JET J7J J7I J/T M JB1 J7I Tfl 

== JLJ-[ J2/J 12/2 2 3 3 4 4 ' 

since the rearrangement of order involves six jumps of a symbol over a 
different symbol, and each jump reverses the sign of the expression by (2-21). 

Hence 



so that E B 2 = - 1. We can verify similarly that E l E 6 = - E b A\ , etc. 

Thus we have five symbols satisfying (2-21). Both equations of (2-21) 
are included in the form 



)=-8^ (^=1,2,3,4,5), (2-23) 

where is the symbol defined in (1'43). 

Any product formed by repeated multiplication of E 19 E 2 , E. 3 , # 4 can be 
reduced to the form E 1 p E 2 Q E^ r E 4L 8 , since in collecting the factors the 
alteration of order can at most change the sign of the product. Also, since 
E fJ ?= 1, E^ J reduces to E^ or 1. Thus, disregarding sign, the product 
reduces to one or other of sixteen forms: 

1, E^ E^E V , E fi E v E a , E^EsEt (^ v, a = 1,2,3,4; p*v* or). 

Multiplying (2-22) by E , we have ( 2 " 24 ) 

iE l E 5 = E 1 2 E 2 E^E^= E 2 E 3 E^, 

so that by using E 5 the triple products can be reduced to double products. 
Disregarding factors 1, i, the forms (2-24) are equivalent to the sixteen 
forms i, E^ E^E V (^ v= 1,2, 3,4,5; ^v). (2-25) 

As here written they are all square roots of 1; since 

(EpErf-E^EpE^ -E^E V E V = -EJE,?--!. 

A linear function of the sixteen expressions (2-25) with algebraic coeffi- 
cients (real or complex) will be called an E-number. We see that the opera- 
tions of addition, subtraction and multiplication applied to JS?-number 
will always yield JB-numbers. In virtue of this property the sixteen expres- 



22 Wave-tensor Calculus [2-2 

sions are said to constitute a complete set. Similarly in algebra the symbols 
1 and i constitute a complete set, since the operations of addition, subtrac- 
tion and multiplication applied to complex numbers always yield complex 
numbers.! 

For reasons which will appear later the sixteenfold complete set here 
introduced is called an orthogonal set. 

The -E-iiumbers are a particular case (n = 4) of Clifford's numbers, J 
which are formed analogously from any even number n of independent 
anticommuting square roots of - 1. Since the J-numbers, or their equi- 
valent matrices, play a fundamental part in the physical theory which we 
shall develop, the theory is dependent on the choice n = 4 which we make at 
the outset. This choice will ultimately be justified in 16*8, where it is 
shown that it is imposed by the epistemologicai principles involved in the 
conception of measurement. 

2* 3 . Notation of the E -symbols . 

We shall write E^E^ (p, v= 1, 2, 3, 4, 5; /^ ). (2-31) 

For uniformity we also give the original five symbols an alternative double- 

suffix notation, viz. jr JT w /OQO\ 

\ = A)/* = ~ A*o ( 2 ' 32 ) 

Then the sixteen expressions (2-25) which constitute the complete set 
become ^ ^ ()Lt)1 , = o, 1,2,3,4,5; p*v). (2-33) 



By (2-31) and (2-32) we have in all cases E^ v = E vfl . In making up the 
complete set of sixteen symbols it is arbitrary whether we employ E^ v or 
E V p . It would, of course, be redundant to include both. 

By using (2*21) and (2-22) we find the following general rules of multi- 
plication: ^^=-1, (2-341) 

*V^a= -E^ v = E va , (2-342) 

E^E^ E^E^iE^ (2-343) 

where /*, v, a, T, A, p is any even permutation of 0, 1, 2, 3, 4, 5. For an odd 
permutation E^E^^ -iE^. 

The summation convention is not used in the above formulae. Unless 
otherwise stated we shall limit the summation convention to the row-aiid- 
column suffixes of matrices and wave vectors. In later developments the 
symbols E^ v will be identified with matrices; they will then have the form 

t Some writers use the term "complete set" for the group of linear expressions (in this 
case the j-numbers). The expressions in (2-25) would then be called "generators" of the 
complete set. 

J Amer. Jvurn. Math. 1, 350 (1878). 

The summation convention is also employed when well-known formulae are quoted from 
general relativity theory. 



2-3] The SixteenfoU Frame 23 

C^/zv)a)3> where a and ]8 indicate the element in the ath column and j8th row 
of the matrix. In that case the summation convention will apply to a and ]3, 
but not to the suffixes p., v which distinguish one matrix of the set from 
another. 

By (2-342) and (2-343) the JE-syrnbols commute or anticommute according 
as they have no suffix or one suffix in common. We therefore obtain a sub-set 
of mutually anticommuting symbols by fixing one of the two suffixes and 
letting the other vary, e.g. 

^30> ^31> ^32> ^34 ^35' 

We call such a sub-set a pentad. There are six different pentads; and each 
symbol is a member of two pentads. Our original symbols E ly E 2 , E$, E 4 , K 5 
constitute the pentad with fixed suffix 0. It will be seen that, if wo start 
from any of the other pentads ami follow the same treatment, we reach the 
same complete set. 

There exist also triads, i.e. sets of three mutually anticommuting E- 
symbols, which do not form parts of pentads, viz. 

E^, E va9 E afA (n*v*a). (2-35) 

The maximum number of mutually commuting ^-symbols (excluding i) 
is three; for no two of them can have a suffix in common, and therefore three 
symbols exhaust the six possible suffixes. The three commuting symbols 

.are accordingly 

E^ , E aT , E^ (/*, v, or, T, A, />, all different). (2-36) 

We call such a set an anti-triad. Adding to it the symbol i, which commutes 
with all symbols, wo obtain an anti-tetratl. 

The two triads E^, E va , E Q ^\ E rX , 7? v E pr , (2-37) 

where p,, v, a, T, A, p are all different, are called conjugate triads. They have 
the property that every member of one triad commutes with every member 
of the other (see 3-8). 

We shall often employ an alternative single-suffix notation for the 
JSr-symbols (2-33), viz. E^ (JLC= 1, 2, ... 16). It is then understood that the 
first five symbols form a pentad, and that 

Ei* = i, (2-38) 

but the order of the others is left unspecified. A general U-number is then 

T^Z t^, (2-39) 

/x-l 

the coefficients ^ being algebraic numbers, real or complex. The individual 
terms t^E^ are called components of the ^/-number. The algebraic com- 
ponent Ji 6 JS7i6i or i 18 , will be called the quarterspur (abbreviated as qs). We 
have therefore qg T ^ ^ ^ (2-395) 

If qs T = 0, I 7 is said to be degenerate. 



24 Wave-tensor Calculus [2-4 

2*4. Linear Independence of the E -symbols. 

(a) If a complete set is multiplied through by any one of its members, we 
obtain the same set in a different order, apart from algebraic factors 1 or 
i. This follows from (2-34). 

(6) If an JS-number vanishes, every component is zero. For suppose that 
the ^-number 



. + t m S m = 0, (2-41) 

the coefficients being non-zero. Multiply through by E m \ it follows from (a) 
that we obtain an expression of the same form and with the same number 
of terms as (2-41). The last term is t m E m 2 = t m (or it m E u ). Accordingly, 
let the result be / 



/ n tv jo\ 

.-t m = 0. (2-42) 

Let E r be one of the symbols which anticommute with J57 a . Multiply (2-42) 
firstly by initial E T and secondly by final E T , and add. Then 

t OL (E T E OL + E 0i E r ) + tp(E T Ep + EpE r ) + ...^2t m E T = 0. (2-43) 
The first term vanishes, and possibly some of tho other terms; but the 
equation cannot wholly disappear since the last term does not vanish. If 
E T commutes with Ep, (E T Ep + EpE r ) = k 2iE a , where E a is another symbol 
of the set, by (2-343). Hence (2-43) reduces to an expression of the same 
form as (2-41) but with fewer terms. 

By repeating the whole process as often as required we remove all terms 
except the last; we are then left with an equation containing just one non- 
zero term which is absurd. Thus an equation of the form (2-41) is impossible 
unless all the coefficients are zero. 

This shows that the ^-symbols are not connected by any linear algebraic 
identity. In other words the set is complete but not redundant. 

2*5. Miscellaneous Properties. 

The following easily established properties of J?-symbols are collected here 
for reference: 

(a) Each symbol (except E IQ ) anticommutes with eight symbols, viz. the 
remaining members of the two pentads to which it belongs. It commutes 
with the remaining eight symbols, which include itself and E u . 

(6) Each symbol (except E 1B ) anticommutes with at least one member of 
any given tetrad. (A tetrad is formed by four members of a pentad.) For if 
the tetrad is E ol , E Q2 , E Q3 , J5 04 , the symbol E^ has one suffix in common with 
one of these unless both a and r are 5. But a and r cannot be the same. 

(c) If an JK-number commutes with E^ , every non-vanishing component 
commutes with E^. For the condition that Et v E y commutes with E^ is 

s,u^-v,)=o. 

Terms for which E V9 E^ commute disappear; terms for which E V9 E^ anti- 
commute reduce to the form 2t v E a by (2-342). No two terms reduce to 



2-6] The Sixteenfold Frame 25 

the same E a . By 2-4 (6) the coefficients of these surviving terms vanish 
separately; that is to say, t v = for those components E v which do not 
commute with E^ . 

(d) Similarly if an -E-number anticommutes with J , every non-vanishing 
component anticommutes with E^ . 

(e) If an jE7-number commutes with each member of a tetrad, it is an 
algebraic number. For by (c) its non-vanishing components commute with 
each member of a tetrad, and by (6) no ^-symbol other than A\ 6 can do this. 
The J5-number therefore reduces to E^t^, or it lB . 

(/) For any /?-number T we have 

ji=lG 

2 E ' TE = -16qsT. (2-51) 

/t=i 

Consider the component t v E v . We have 

EpE Ep = -E v , if E v , A; commute, 

= + E v , if E v , Ep onticommute. 
Hence, if v^ 16, we have by (a) 



For v= 16, ^A;^ 16 ^= 

Hence ^E^TE^ - 16# 16 16 = - I6qs T. 

(g) The coefficients ^ of an ^-number satisfy 

tp= ~qs(T^)= -qs(^y). (2-52) 

Let H=TE fJL . Each component of S corresponds to a single component of 
T by 2-4 (a). The quarterspur of 8 corresponds to the component t^ E fl of T, 
and is therefore equal to t fA E fJi .E^= t^, which proves the theorem. 
Combining (2-51) and (2*52) we obtain 

m^-l^^(TE a )^^E^TE E^^E TE^ (2-53) 
(h) If S and T are U-numbers 

(2-54) 



2*6. Reciprocals. 

Let S and T be Jff-numbers. Generally there exist two quotients T/S, 
which are the ^-numbers R, R' defined respectively by 

R8=T, SR' = T. (2-61) 

Considering the first of these equations, the vanishing of the jB-number 
RST requires that every component of it should vanish. We have there- 
fore 16 equations (linear in r^) to determine the 16 coefficients r^ of the 
J5-number R. A solution will exist unless the determinant of the coefficients 



26 Wave-tensor Calculus [2-6 

of the r^ vanishes. Since the coefficients of the r^ are furnished by 8, the 
existence or non-existence of a solution depends on 8 but not on T (assuming 



If there is no solution, i.e. if 8 fails as a divisor, 8 is said to be singular. 

In particular, taking jf=l, an JEJ-number 8 will have a reciprocal R 
(such that R8 = 1) unless it is singular. A singular J?-number has no re- 
ciprocal. 

If R8=l, SR' = 1, (2-62) 

we have R = R (8R f ) = (RS) R = R', (2-63) 

so that the same reciprocal is obtained by either definition. An jE-number 
commutes with its reciprocal. 

Let T be an JSJ-number which commutes with 8. Denoting the reciprocal 
of S by S~\ we have S -i ^ T g ^ S -i _ $-1 ^ ST ^ g-^ 

whence S~ 1 T=TS- 1 . 

So that an U-number which commutes with 8 commutes also with its 
reciprocal. We can also show that, if two TiJ-numbers commute, their reci- 
procals (if any) commute. 

If 8 is singular, the vanishing of the determinant of the coefficients makes 
it possible to obtain an infinitude of solutions of 

RS = 0. (2-64) 

Any such solution R is called a jtseudo-reciprocal of 8. A pseudo-reciprocal 
is necessarily singular. If R is a pseudo-reciprocal of 8, XR is also a pseudo- 
reciprocal of 8, X being any jG7-number ; for if RS = 0, X RS == 0. A product 
of JB'-numbers is singular if any of its factors are singular. 

It is important to notice that, when 8 and T are j&-numbers, the equation 
ST = does not imply that either $ = or T = 0. There is an alternative, 
viz. that 8 and T are singular. If, however, 

SEpT = Q (2-65) 

for every symbol E^ of the complete set, then either $ = or T = 0. For 
suppose that 8^0. Then by (2-53) 



= 0, by (2-65). 

Hence t a = 0. Since this applies to every component t a , it follows that T = 0. 

If 8 is singular and RS contains no non-algebraic terms, then 

jRS = 0. (2-66) 

For if RS were a non-vanishing algebraic quantity a, JR/a would be the 
reciprocal of 8, so that 8 could not be singular. 

Since E^.E^l, the symbols E^ are not singular. Hence if 



2-7] The Sixteen/old Frame 21 

2*7. Transformation of Complete Orthogonal Sets.f 
Let F^qE^q', (2-71) 

where 92 / = ?'?= 1 - (2-72) 

Then the F^ form a complete set having the same structure as the set of E^ . 

This is proved by showing that the relations (2-34) pass over unchanged 
from the E^ to the F^. Taking, for example, (2-342) 

F^F^qE^q' .qE^q' 

^qE^E^q' by (2-72) 
= qE va q' = F v <,. 

Here q and q' may be jE-numbers or they may involve entirely new symbols. 
We make no assumption as to their nature. 

The converse theorem is that if JE^, F^ are two complete orthogonal sets, 
arranged in corresponding order so that F^ , F v commute or aiiticomniute 
according as E^ E v commute or anticommute, there exists a transformation 
(2-71) connecting them. We shall prove this under the restriction that 

(a) The F^ are ^-numbers, or 

(6) The Fp are new symbols which commute with all the E^ . 

16 16 

Let P-aZJ^L, P'-aSlZUP, (2-73) 

l r r l r f* 

a being an algebraic number. Consider the expression F V PE V . It has 16 
terms of the form aj p / p jg \ ^ 

If E v , Ep anticommute, and therefore F v , F^ anticommute, this becomes 
-oF^J^JEf,,, which is of the form -*F Q E a by (2-342). If E v , E^ and 
F V9 Fp commute, it becomes ^.F^F V E^E V , which is of the form a (iF a ) (iE a ) or 
-u.F E a by (2-343). Thus F V PE gives the 16 terms of -P in a different 
order; hence ^ pE ^ = _ p (2 . 741) 

Similarly E V P'F V =-P'. (2-742) 

Multiplying by final E v and final F V9 respectively, these give 

F V P = PE V , E V P' = P'F V . (2-75) 

Case (a). F^ is an jB-number. 

Multiplying together the two equations in (2-75), we have 

E V P'PE V =P'F V F V P= -P'P. 
Hence, multiplying initially by E V9 



Therefore P'P commutes with every E V9 i.e. with every symbol in the 

calculus. It is therefore an algebraic number. Reserving the singular case 

t This transformation was introduced by G. Temple, Proc. Boy. Soc. A, 127, 342 (1930). 



28 Wave-tensor Calculus [2-7 

P'P = 0, for consideration in 2-8, we can choose a so as to make P'P = 1. Tt 
follows that PP' = 1; and by (2-75) 



so that P is the required transformation operator q in (2-71). 

Case (b). F^ commutes with the E^. 

Then P = P'. If P is multiplied by E^F 19 we obtain the same 16 terms in 
a different order, except that those which commute with E l (and therefore 
with FJ acquire a factor i 2 by (2-343). Thus 8 terms are reversed in sign. In 
the product P x P, each term of P occurs 16 times, 8 times with the original 
sign and 8 times with reversed sign, except that E IB F 19 occurs 16 times with 
reversed sign. Hence p 2 = a 2 ( _ 16^ 16 P 16 ) = 16a 2 . 

Taking a = J, we have PP' = P 2 = 1. Then by (2-75) 



Hence the required transformation operator is 

? = <?' = iS# /t *;. (2-76) 

2-8. The Singular Case. 

In Case (a), but not in Case (6), it may happen that PP' = and the fore- 
going method of determining q breaks down. We shall show that never- 
theless there is a transformation F^qE^q'; but instead ofq = P, we have 
^P^), where p^^^^ p^^^E^F^, (2-81) 

and E a is one of JK-symbols. The transformation previously given corre- 
sponds to a = 16; if that fails we try another value of <r, until we find one such 



Ordinarily the vanishing of PI " does not imply that either P or P' is 
zero; but in the present case we have, by (2-75), 



for every E v . Hence by (2-65) either P = or P' = 0. 

Let EM = (iE a ) Ep (iE a ) --E^E,,. (2-82) 

Since this is a transformation of the form qE^q' with qq' = 1, the JS (or) form 
a complete set. By (2-82) E^^E^ or -E^ according as E a commutes or 
anticommutes with E^\ thus the transformation simply reverses the signs 
of eight members of the set. We call the set E^ a reflection of the set E . 
Including the original set (reproduced when a = 1 6) there are sixteen different 
reflections, which correspond to the sixteen possible combinations of sign 
in an initial tetrad E l9 E 2 , J 3 , E. 

By (2-51) S a ^)=16qsJ^. (2-83) 



2-9] The Sixteen/old Frame 29 

Hence by (2-81) 

E a PMJ0 a = - IGaZ^qs^ - 16o^ 16 # lfl = 16a, (2-84) 

since qs E^ = unless p, = 1C. Hence at least one of the quantities 1* V} E has 
a non-vanishing quarterspur. Therefore at least one of the quantities P (a) 
does not vanish. 

The quarterspurs of P (o) E a and E a P' (af) are equal; for they are the quarter- 
spurs of - ocS2^ /y a > and - oEE^F^ , which are equal by (2-54). Hence, in 
securing that P^^^Q, we also secure that P' (a) ^ 0. 

Having found non-vanishing P^ a \ P r(a \ we use these instead of P 9 P', and 
repeat our previous analysis as far as (2-75), obtaining 

f w pw=pto]B, 9 E V P'M = P'WF V . (2-85) 

It will be found that E a remains passive in the middle of the expressions 
F v PWE v and does not affect the argument. Proceeding to Case (a), we find 
as before that P'^pw is algebraic. Further, it cannot vanish; for, as shown 
for P, P' at the beginning of this section, its vanishing would require that 
either P^> or P' (a) is zero. 

It may seem curious that we should be able to choose a arbitrarily in the 
transformation jF^ = P^/S^ P' (CT) . The explanation is that by changing a we 
introduce a purely algebraic factor which is absorbed in a. Evidently the 
transformation could be further generalised by substituting an arbitrary 
^/-number in place of E a . 

We notice for future reference that there is at least one reflection E^ 
which is connected with F^ by the unmodified transformation q = c 
q' = KZE^FH . For, choosing o- so that P< a) , P'< a) * 0, we have 



-aEE.EpE.Fn~ - 

2-9. Application to Relativity. 

The transformation F^qE^q' is formally the same as the transformation 
(1-463) of a mixed tensor T' = qTq'. Limiting ourselves for the present to 
Case (a), q is now a non-singular jfir -number instead of a non-singular matrix. 
We shall find in Chapter in that fourfold matrices are a special representa- 
tion of Jfi/-numbers. Thus it is appropriate to generalise the definitions of 
tensors in Chapter I by substituting symbolic JS?-numbers for matrices. 

In physical applications we shall call a "complete set of E^ a symbolic 
frame. By the above transformation we obtain a different but equivalent 
symbolic frame F^ . If the E^ are taken to be mixed wave tensors, the 
change from one symbolic frame to an equivalent frame is a tensor trans- 
formation. A change of symbolic frame is then part of a general change of 
system of description, and other quantities occurring in the description are 



30 Wave-tensor Calculus [2-9 

changed simultaneously according to the wave tensor character assigned 
to them in the group of descriptions contemplated. 

The present analytical theory is being developed to serve as a tool in 
physical investigations, and we cannot pledge ourselves always to use the 
tool in one particular way. But the primary application will be that certain 
characteristics of a physical system are described by JP-numbers which are 
invariant for changes of the system of description, and therefore correspond 
in conception to an absolute structure transcending our variable description. 
Let T = T&tpEp be one of these invariant ^-numbers. The invariance requires 
that when we transform to a new symbolic frame E^', the coefficients t are 
changed to */ so that T^E^S^'E^. (2-91) 

The arrays ^ , t^ are regarded as components of the same physical entity 
referred to two different reference frames E E'. 



It is convenient to call the transformation E^E ' a rotation of the 
symbolic frame. Rotation is here given a somewhat generalised meaning; 
the emphasis is on the fact that it is a type of change which does not involve 
any intrinsic distortion of the frame. The frame JB ' has the same intrinsic 
structure as the frame E^, namely that expressed by equations (2-34). 

Having defined rotations of the frame we can now define corresponding 
rotations of a physical system described by ^-numbers. Consider a system 
described by invariant A 7 -numbers Z^S^JS^, J7 = S^jB r /x , etc. Let the 
system undergo a change such that T-+T', U-> U f , etc., where 

T'^pEp', U' = Vu^. (2-92) 

Then the new physical system is constructed in the frame E^ according to 
the same specification as that by which the original system was constructed 
in the frame E^. In other words the system has rotated with the frame. 
Clearly the systems (T, C7, ...) and (T f , U', ...) have the same kind of 
equivalence as the frames. They are intrinsically similar, as the frames are 
intrinsically similar. 

Normally the rotation of a physical system is described by referring it to 
a fixed frame. We therefore require the components of T' in the original 
frame E^. Denoting these components by t^ (not the same t^ as in (2-91)) 
the condition is r = a ^, = a ^ (2 . 93) 

The transformation J /4 ->< / / represents a rotation of the physical system 
relative to the fixed frame E^. Since E^^qE^q'y we have by (2*93) 

Xtp'E^q&tpEJq'. (2-94) 

The nature of the transformations of t^ determined by (2-94) will be studied 
in detail in Chapter iv. 

Any change ^->^' represents some imaginable change of the physical 
system described by ^ . The peculiarity of the transformations which satisfy 



2-9] The Sixteen/old Frame 31 

(2-94) is that the new system is intrinsically similar to the old, and the 
change is therefore pictured geometrically as a rotation without distortion. 
More generally we call such a change a relativity transformation. It can bo 
detected (if at all) by observing relations to extraneous physical objects 
that do not form part of the system to which the relativity transformation 
is applied. 

The relativity of our orthogonal symbolic frames is precisely analogous 
to the relativity of Galilean frames of space and time. Space-time frames 
are all alike initially. If we speak of a frame x, y, z, t, it is impossible to define 
in an absolute way which frame, out of an infinite number of equivalent 
frames, we refer to. But when once we have selected and labelled an initial 
frame A, any other frame B can be defined relatively to it by specifying the 
space rotation and Lorentz transformation which would convert A into B. 
Similarly we cannot define in an absolute way the frame E^ which we select 
initially. But any other frame E^ can be defined relatively to E^ by speci- 
fying the transformation symbol q (which is an ^-number of the form 
Sgr^ .By) connecting E^ and E^ f . We use these symbolic frames as the basis of 
a relativity theory which (we shall find) includes, but is somewhat more 
comprehensive than, the relativity of Galilean frames of space and time. 

Attention may be called to the perfect adaptation of the mathematical 
symbolism to the physical conditions. Owing to relativity we are unable to 
define in an absolute way the physical frame initially selected, which we label 
Ep . It is therefore appropriate that we should be equally unable to define 
in an absolute way the label E^ which we affix. For the set of symbols E^ 
is only defined by its structural properties (2-34), and these apply equally 
to Ep or to any other complete orthogonal set. The complete physical 
equivalence is therefore represented by a complete mathematical equi- 
valence. We lose this perfect adaptation when we use special kinds of / /A , 
e.g. matrices. 

There is no absolute distinction between a rotation of the physical system 
and a rotation of the frame in the opposite direction; and in elementary 
theory the term "relativity rotation" is applied indifferently to rotations 
of the physical system and of the frame. But after the first results of this 
equivalence have been gleaned, there is seldom anything to be learned by 
introducing rotations of the frame. If the frame is rotated, we have to 
transform simultaneously the specification of all objects, fields, boundary 
conditions (including boundary conditions at " infinity"), normalising 
conditions, etc., concerned in the problem. On the other hand, keeping the 
frame fixed, we can introduce relativity rotations of a particular object, 
leaving the other objects concerned in the system unchanged. In order that 
it may possess independent relativity rotations, the object must be con- 
ceived as separable from the rest of the system contemplated. A separable 



32 Wave-tensor Calculus [2-9 

object will have a structure described by invariant Jff-numbers T, Z7, ..., so 
that transformations of the form (2-94) represent displacements of the 
object without intrinsic change of its structure. 

Thus in later developments we are concerned with independent relativity 
rotations of individual objects, which provide much wider scope for the 
application of relativistic principles than a rotation of their common frame.! 
For this reason, "relativity transformation" will normally mean displace- 
ment without intrinsic change of an object referred to a fixed frame, though 
it may also be applied to a rotation of the frame if occasion arises. 

In practice an object cannot be rigorously separated from its surroundings ; 
if it could be separated, it would not be accessible to observation. But that 
does not do away with the usefulness of the conception of a separable object. 
One of the greatest achievements of current quantum theory is that it 
has found a rigorous method of avoiding this dilemma. An incompletely 
separated object is represented as a probability distribution over completely 
separated states. The environment then affects, not the state, but the 
probability attached to the state. We may therefore, with all rigour, apply 
relativity transformations to the IS-numbers describing the states pro- 
vided, of course, that the states are such as can be specified by invariant 
.E-iiumbers. This last reservation is liable to be overlooked ; and ^-numbers 
(or the equivalent Dirac wave functions) have often been applied to states 
which obviously do not possess the relativistic properties which jE-numbers 
are designed to represent. 

From one point of view the assumption that there exist in nature equi- 
valent 16-fold frames, which can therefore be appropriately represented by 
equivalent sets of j&'-symbols, is a hypothesis the fundamental hypothesis 
of our theory. But actually we appeal to an epistcmological principle which 
goes deeper than that. We will call it the k< Principle of the Blank Sheet". 

Physics is concerned with the problem of distinguishing and classifying 
the distinctions of objects, states, events. Exact measurement is a process 
of determining and classifying minute distinctions. To develop a theory of 
the characteristics which can be distinguished and of the measurement of 
the distinction, we require a blank sheet to write on not a sheet already 
scribbled over with vaguely recognised distinctions. A group of intrinsically 
indistinguishable frames is chosen as the basis of a description of the uni- 
verse, in order that the theory of distinguishable or measurable phenomena 
to be erected on it may go down to the very origin of their distinction. Tn 

t The above remarks refer to the more usual problems of quantum theory in which a 
number of objects are referred to a single frame. An important Intermediate stop is the 
consideration of the relativity rotations of two objects referred to a double frame E fJL F v . 
In this case any combination of rotations of the two objects is equivalent to a transformation 
to an equivalent double frame E^F f . This (rather unusual) development is of great 
importance in the special problems treated in the present book. 



2-9] The Sixteen/old Frame 33 

practice we do distinguish frames of space-time otherwise than by their 
transformation relations to one another; we distinguish them as being at 
rest relatively to the earth, sun, etc., or as having a special orientation with 
respect to the earth's gravitational field; but we conceive frames of space- 
time as initially indistinguishable in order that these distinctions may be 
properly inserted in the development of the theory and not hidden in its 
initial assumptions. We do not assert that there exist intrinsically indis- 
tinguishable 16-fold frames of reference in the physical world; it is only for 
an ideally simplified universe that this would be true. Our principle is that 
such distinguishability of the frames as occurs must be treated as a positive 
characteristic to be represented by appropriate symbols and combined in a 
unified theory with the other distinctions studied in physics. To exhibit a 
positive characteristic, we have to imagine a frame which initially lacks it. 

Our mode of thought requires us to formulate some kind of frame or 
background for physical phenomena. Let the background be a white sheet 
to show up the phenomena, not a jazz-painted camouflage against which 
they may lie undetected. 

Case (6), in which the equivalent frames E^ , F^ consist of entirely different 
symbols, has also an important physical application. Since q is now a 
mixture of J5-symbols and J-symbols, the relation between the two frames 
is not describable by reference to the A'-frame only. If we have three such 
frames E, F, G, the transformations y Ey ,q Ei } cannot be compared, and there 
is no meaning in saying that the change from E to F is greater or less than 
the change from E to G. Equivalent frames of this kind are required when 
we deal with the properties of two similar atoms or two electrons, con- 
ceived as non-interacting. Two electrons are intrinsically similar (or equi- 
valent) but are not the same; we cannot specify different degrees or different 
kinds of not-the-sameness, as we do for equivalent space-time frames. If 
there is interaction the case is somewhat altered, and the electrons are not 
so definitely distinct; but here again the Principle of the Blank Sheet 
requires us to start with frames corresponding to non-interaction, into 
which interaction is introduced as an explicit perturbation. 



CHAPTER III 
THE RESOLUTION OF MATRICES 

3* 1 . Four -point Matrices . 

We are now going to show that fourfold matrices may be expressed as 
2?-numbers ; so that the theory developed in Chapter n has a particular 
application to matrices. 

First consider the six matrices 

a = 100 8p = Q 010 #y = 001 

1000 0001 0010 

0001 1000 0100 

0010 0100 1000 

D a =l 000 D0=l 000 JD y =l 000 

0100 0-100 0-100 

00-10 0010 00-10 

000-1 000-1 0001 

The system of nomenclature is that the suffixes a, /J, y have reference to the 
three ways of pairing four numbers, viz. 12, 34; 13, 24; 14, 23. 

We also introduce two alternative notations for the unit matrix, viz. 

S 8 = J9 S =1. 
The following results of matrix multiplication are easily verified: 

flfy = fl y , D^D^D V \ (3-111) 

8 u *=i, # a 2 =i; (3-112) 

S a D a = D a ,S a , S a ^=-Z>0AS a ; (3-113) 

with similar results obtained by permuting a, /?, y, but not S. 

The commutative properties may be summarised as follows (a, 6 = a, /?, y, 8) : 

S a S b = S b S a , D a D b = D b D a9 S a D b = (ab)D b S a , (3-12) 
where (a&) = (6a) = l if a = 8, or 6 = 8, or a = 



= 1 otherwise ( " 

The product of any number of these matrices in any order can be reduced 
to one of the sixteen forms: 

S a D b (a,6 = a,j8, y ,8). 

For we can bring all the $'s to the beginning and the >'s to the end by 
applying (3-12), and then reduce the 's to a single S and the D's to a single 
D by applying (3-112) and (3-111). If either the S factor or the D factor 
disappears, we insert the unit matrix S s or JDg to preserve homogeneity. 



3-2] The Resolution of Matrices 35 

Thus the sixteen forms constitute a complete set in the sense explained in 
2*2. If we call a linear function of them with algebraic coefficients an SD- 
number, the operations of addition, subtraction and multiplication applied 
to D-numbers will always yield /SZ)-numbers. 

3-2. Pentads. 

It follows from (3-12) that 

(8 a D b ) (8 c D d ) - (be) (ad) (S c D d ) (S a D b ). (3-21) 

Hence the condition that S a D b and 8 D d anticommute is 

(bc)(ad)=-l. (3-22) 

Let us write down the matrices S c D d which anticommute with /S a /> 8 . 
Here 6 = 8, so that (6c) = l. Hence (orf) = -1; and since a = a, we have 
d = ]8 or y. The suffix c can have any value. Hence the matrices are 

Sfy> fyDp, S ? Dp, SsDp, S a D y , ty D y , S y D Y , S^D y . (3-23) 

Selecting one of these, S 8 Dp, we find in a similar way that the following 
anticommute with it: 

S a D a , 8 a fy, S a D y , 8^, S^, S Y Dp, 8 y D yt S r D 8 . (3-24) 
Hence the following anticommute both with $ a />a and 8$ Da: 

S^Df), 8^, S y Dp, S Y D Y . (3-25) 

The first of these is the product of /S a jD s and 8$ Dp. A symbol which anti- 
commutes with two symbols necessarily commutes with their product; thus 
no further matrices can anticommute with the triad: 

S a D 8 , S B Dp, 8^. (3-26) 

It will be found that the remaining three matrices in (3-25) anticommute 
with each other, so that 



constitute a pentad of mutually anticommuting matrices. 

Dropping the superfluous 8$ and 1 D 8 , and inserting a factor i where 
necessary to make the square of the matrix equal to - 1, the pentad is 

AS af iDp, iS y D y , S a Z) y , S Y D ft . (3-27) 

These five matrices accordingly satisfy the same conditions (2-23) as a 
pentad of JS-symbols, and constitute a particular identification of E l9 J 2 , 
j 3 , ^ 4 , ES . If identified in this order they are found to satisfy (2-22). All the 
theorems of Chapter H then have an application to matrices. 

The complete set E^ can accordingly be identified with the complete set 
i (ab)* 8 a D b , the factor i being inserted when (ab) = -h 1 in order to make the 
sauare eaual to - 1. 



36 Wave-tensor Calculus [3-2 

The five other pentads can be found from the theory of JE-symbols, or 
more simply by permuting a, ]8, y in (3-27). 

A set of four anticoinmuting four-point matrices was first introduced 
into physical theory by P. A. M. Dirac in his wave equation of an electron. 
The particular matrices used by Dirac form part of one of the pentads here 
found. It was shown by J. v. Neumann that the complete set consisted 
of 16 matrices.! The complete set S a D b was first studied in this connection 
by the author. J It has been pointed out by P. du Val that a similar analysis 
had been developed in connection with the theory of Kummer's Quartic 
Surface. 

3*3. Components of Matrices. 

In 3-2 we have found a representation of the U-symbols, and hence of all 
JB-numbers, by fourfold matrices. We shall now prove the converse, viz. 
that every fourfold matrix will represent an ^-number; that is to say, any 
fourfold matrix T can be expressed in the form 

T^t^, (3-311) 

i r r 

where E l9 E 2 , E 3 , E, E 5 are the matrices (3-27). 

The meaning of (3-31 1) will be clearer if we insert the row-and-column 
suffixes a, /? of the matrices, viz. 

T^ = S^(^) aj3 . (3-312) 

Considering in succession the 16 combinations of suffixes a, j3, we have 16 
equations to determine the 16 algebraic coefficients t^ . The values of t^ are 
unique; for if there were another set of values t^, we should have by sub- 

traction 



Hence by 2-4 (6), ^-y = 0. 

As we shall presently solve these equations for t^, it is not necessary to 
stop to prove here that the condition for the existence of a solution (non- 
vanishing of the determinant of the coefficients) is satisfied. 

We call 1^ (or E^) a component of J 7 , and p a matrix suffix, as distin- 
guished from the elements T a p and the row-and-column suffixes a, /?. 

The sum of the diagonal elements of a matrix is called the spur, and will 
be denoted by {T}. 

The matrices S a D b have no diagonal elements unless a = 8. Also we see 
from the definitions of D a , JCjj, D y in 3-1 that their spurs vanish. Hence 

t Zeits.fur Physik, 48, 881 (1028). 
J Proc. Hoy. 8oc. A, 121, 524 (1928). 

The connection with the Kummer collineation group has been treated fully by 
0. Zariski, Amer. Journ. Math. 54, 466 (1932). 



3-3] The Resolution of Matrices 37 

{S a D b ] = 0, except {S 8 D B } = 4. Changing to the E^ notation, we have 

{^} = for ^=1,2,. ..15, {# 16 } = 4i. (3-32) 

Taking the spur of (3-311), we have 

T (3-33) 



by (2-395). Thus the spur of a matrix is (appropriately) four times the 
quarterspur. It is to be remembered that for a general symbolic Jr-number 
(not identified with a matrix) the diagonal sum would have no meaning. It 
is for that reason that we have introduced the quarterspur as a more general 
characteristic, not implying matrix representation. 



By (2-52) tp= -qs(^rH - H^Z 1 } ( 3 ' 34 ) 

by (3-33). We have thus an explicit formula for the components t of a 
matrix. 

Owing to the great importance of (3-34) it may be desirable to give a 
direct proof. Multiply both sides of (3-311) by E v , and take the diagonal 
sum; we have 



Now E v Ep reduces to a single symbol, whose spur vanishes by (3-32) except 
, = v. Hence , . ,. 



which is equivalent to (3-34). 

We can write (3-34) in a form which avoids the use of the symbol { }. 
Inserting row-and-column suffixes (and temporarily dropping the matrix 



Divide the symbol T into two portions each carrying a suffix, thus, 



Since the suffixes are explicitly indicated, we may rearrange the order of 
the factors, 



the suffixes being omitted after the rearrangement since they follow the 
chain rule. Thus (3-34) becomes 



(3-35) 
We call T! and T symbolic factors of T. 

In particular, if a matrix J is the outer product of two vectors 0, #*, so 
that 

J=VX*> (3-36) 

the components are . T * w / 

F -* (3-37) 



38 Wave-tensor Calculus [3-3 

The expectation value of an operator X with'respect to wave vectors 
and x* is defined to bef x = x*X0 H- x *^. (3-38) 

It follows from (3-37) that the expectation value of E^ is y^/fa. 

3*4. General Orthogonal Frames. 

The SD matrices which constitute the symbolic frame used in 3-3 are of a 
special type, called four-point matrices, since they have only four non- 
vanishing elements. By 2-7, Case (a), an equivalent frame E^' is obtained 
by the transformation jy _ ^ , ^ = ^ ^ 3 . 41 j 

where g is any non-singular JS7-number, and therefore in the present applica- 
tion any non-singular fourfold matrix. 

The matrices E^ of the new frame will not generally be four-point matrices. 
We have therefore to consider whether the results of 3-3 will apply to 
the new frame. 

The spur {E^} is invariant for the transformation (3-41). For 



yy = {E^ (3-42) 

since q'q=l. Hence the formulae (3-32) apply equally to the frame E^. 
Except in calculating the spur, no use was made of the special properties of 
SD matrices; and therefore all the results in 3-3 apply to E^'. 

In particular the formulae (3-35) and (3-37) for the components apply to 
any orthogonal frame of matrices. 

Up to the end of Chapter vi we shall (unless otherwise stated) take the E^ 
to be general fourfold matrices which satisfy the conditions for a complete 
set. We shall not specify the particular set of matrices used. This is in accord- 
ance with the relativity principle in 2-9, that there can be no absolute 
description of the reference frame initially chosen; but if other frames are 
subsequently introduced they can be defined relatively to the first frame by 
stating the components (in the first frame) of the transformation matrix q. 
The frame S a D b has served its purpose in enabling us to construct the whole 
set of equivalent frames; but it has no special significance in physics, since 
the structure of the commutation relations is common to all the frames. 
For example, we must not think of 8 a D b as being physically distinguished 
from other legitimate frames in the way that Galilean coordinates are 
distinguished from other legitimate coordinates. Its apparent distinctive- 
ness (shown in the simplicity of the matrices) is really a misfit between the 

f This is a somewhat generalised definition of expectation value. In current theory the 
term is restricted to an expectation value with respect to one wave vector #; x is then replaced 
in the formula by the complex conjugate of 0. It must not be assumed that the familiar 
properties (e.g. that the expectation value is intermediate between the greatest and least 
eigenvalues) hold for the generalised definition. 



3-4] The Resolution of Matrices 39 

physical structure and the mathematical expression of it by matrices. If we 
keep to general symbolic J?-numbers no such misfit occurs, and in that 
respect they give a closer representation of the actualities of physics than 
the matrix representation does. 

It may be asked, What do we gain by introducing matrices instead of 
general JE-numbers? Ultimately I think we gain nothing. I do not think that 
there is anything in the physical constitution of the systems to which we 
apply this calculus that is represented in the matrices and unrepresented 
in the general ^-numbers. The main justification for using a particular 
representation is that it simplifies the algebra in practical problems. Thus in 
Einstein's theory we introduce special coordinates for the discussion of the 
phenomena of the solar system, since the analysis of these phenomena would 
be intolerably difficult if we retained general coordinates throughout. We 
shall sometimes use the frame of four-point matrices in this way to establish 
results known to be invariant, which it is therefore sufficient to prove in any 
one frame of reference. On the other hand we are liable to lose valuable 
insight by premature introduction of special frames or special coordinates. 
Temple has shown that, even in so special a problem as the determination 
of the energy levels of the hydrogen atom, matrices are not required, and 
the work can be carried out with general ^-symbols; his determination 
appears to me not only more illuminating but actually much simpler as 
regards algebraic calculation than the proofs previously given in terms of 
matrices ( 9-3). In any case the use of matrix representation expressly for 
the purpose of facilitating calculation is a very different matter from its use 
in the formulation of the fundamental laws of physics. 

But, whatever the ideal course, I am here limited by the fact that I do 
not propose to reinvestigate the whole quantum theory. I must develop the 
present relativity theory up to a point at which it meets the accepted results 
of quantum theory which are soundly (if unaesthetically) established. 
These results are given in matrix representation by Dirac and others, and 
the conventional nomenclature and definitions have reference to the matrix 
representation. I must have an eye on the theory that I am steering to meet 
before I actually make contact with it; therefore it seems unwise to post- 
pone the transition to matrix representation for long. Meanwhile the 
knowledge that there is an equivalent theory in terms of general symbols 
is reassuring; for I cannot believe that anything so ugly as the multiplication 
of matrices is an essential part of the scheme of nature. 

In 2-7, Case (6), we may take the E^ to be matrices and the F to be 
general symbols, or vice versa ; then q will be a mixture of matrices and general 
symbols. Thus the use of matrix representation does not entirely cut us off 
from general symbols; the gap can be bridged by an ordinary tensor trans- 
formation. I shall be talking chiefly about matrix frames; but if you will 



40 Wave-tensor Calculus [3-4 

inscribe q on the front cover of the book and q' on the back cover then I 
am talking about general symbolic frames ! 

There are two kinds of property which at first sight seem to be expressed 
more simply in matrix calculus than in general symbolic calculus: 

(1) A matrix can or cannot be resolved into two factors. We shall call a 
factorisable matrix a pure matrix. Thus we can recognise a distinction 
between pure and impure matrices, which is not apparent in the corre- 
sponding general symbols; the " factors" of a general U-number are an 
undefined conception. But purity of a matrix is an invariant property for all 
wave tensor transformations, since the two factors (vectors) transform 
separately. There is therefore some invariant characteristic of an JE?-number 
which corresponds to the factorisability of all its matrix representations. 
This characteristic is found to be idempotency ( 5-6). Purity is expressed 
quite as easily by idempotency in symbolic calculus as by factorisability 
in matrix calculus. 

(2) In the specimen pentad (3-27) three matrices are imaginary and two 
are real. This partition persists in all pentads ( 3-5); and it is of great im- 
portance in physics, being the foundation of the distinction between space 
and time. To ascribe real or imaginary character to general symbols would 
involve something not expressible in terms of their commutability relations. 
It is to be remembered that the E^ are all square roots of - 1, whether they 
are represented by real or imaginary matrices. In the case of four-point 
matrices the imaginary matrices are symmetrical and the real matrices 
antisymmetrical (for interchange of rows and columns), and the distinction 
can be equivalently described by reference to symmetry; but the property 
of symmetry or antisymmetry is not invariant for tensor transformations. 
Matrix representation seems to afford the easiest way of expressing this 
distinction; but there would be no great difficulty in working out an alter- 
native treatment by general symbolic methods if desired. 

From Chapter vn onwards our point of view changes, and we shall gener- 
ally restrict the E^ to four-point matrices (SD matrices). That is because 
we have finished contemplating the "blank sheet" and are beginning to 
write something on it; and the property of symmetry or antisymmetry of 
the matrices of a certain frame is one of the first things that we write. 

3*5. Real and Imaginary Matrices. 

A matrix is said to be real if all its elements are real, and imaginary if all its 
elements are imaginary. If any of the elements are complex, or if some are 
real and some imaginary, the matrix is said to be complex. We shall show 
that, if complex matrices are excluded, three members of a pentad are 
imaginary and two are real. 
We first prove by a reductio ad absurdum that five imaginary matrices 



3-5] The Resolution of Matrices 41 

cannot form a pentad. Suppose then that jP lf F^F^F^ F b are imaginary 
matrices forming a pentad; this is connected by a tensor transformation 
with the known pentad (3-27) containing three imaginary matrices E l9 &>, 
E 3 and two real matrices J57 4 , JE^. By 2-7 the transformation connecting 
complete sets E^ , F^ is 

F^PE^P', P-o^^, P' = aS^, PP'-l. (3-51) 

The singular case is avoided by using an appropriate reflection of the pentad 
(3-27); for, as shown at the end of 2-8, there is at least one reflection which 
gives non-zero P and P'. 

Write P=R + iS, P' = R f + i8' 9 (3-52) 

where R, R', 8, 8' are real matrices. Then, since P'P=* 1, 

R'R-ti'S=l, R'S + S'R = U. (3-53) 

By (2-75) F l P = PE ly J^P'-P'I^. 

Hence, separating the real and imaginary parts, 



so that EI R f RE l = R'F^\ R=-R'R. 

Hence E^ R'R = R'RE^ . Similarly R' R commutes with E 2 and E. 3 . Therefore 
by 2-5 (c) it consists of components which commute with E l9 E 2 , E.^. This 
restricts it to the form R'R = a + W5 45 . (3-54) 

Since J? 45 is a real matrix, a and b are real coefficients. 

Again, separating the real and imaginary parts of F P = P/? 4 ,E^P f = P't\ , 
we have 



so that # 4 R'SEi = S'F F^ R=-S'R= R'S 

by (3-53). Therefore R8 anticommutes with # 4 . We can show similarly 
that it anticommutes with E u , jE 24 , ? 34 . This restricts it to the form 

R'S = cEt 5 =-S'R, (3-55) 

where c is real. 

By (3-54) and (3-55) 

R'P = R (R + iS) = a + bE^ + icE^ , 



Therefore 

R'R= R'PP'R = (a + 6^ 45 ) 2 + c 2 ^ 45 2 = a 2 - 6 2 - c 2 + 2a6 J fi? 45 . (3-56) 
Comparing with (3'54), we have 

a = , a 2 6 2 c 2 = a, 

so that 6 2 + c 2 = J, which is impossible since 6 and c are real. 

We can show similarly that a pentad of four real matrices F 19 F 2 , F^, F 6 



42 Wave-tensor Calculus [3-5 

and one imaginary matrix F 3 leads to a contradiction. For, denoting the real 
matrices in the standard pentad by E l9 E 2 and the imaginary matrices by 
E 3 , JK 4 , E & , we have again two matrices -F 4 , F 6 whose character (real or 
imaginary) is opposite to that of the corresponding matrices E^, E 5 , and 
the proof applies without alteration. 

No other case arises, since by (2-22) the number of imaginary matrices in 
a pentad is necessarily odd. Thus the only possible partition of matrices in 
a pentad is three imaginary and two real. 

The theorem has been generalised to matrices of m rows and columns by 
M. H. A. Newman.f If m = 2p, where p is odd, the maximum number of 
matrices in an anticommuting set is 2q+ 1; and of these q+ 1 are imaginary 
and q real. 

A case that might possibly be of physical interest is w = 16. The maximal 
anticommuting sets are then nonads with five imaginary and four real 
matrices. A nonad can be constructed as follows: The sixteen rows are 
designated by double suffixes ocj8 (a, /?=!, 2, 3, 4). Then if E^ denotes a 
4-rowed matrix correlated to the first suffix, and F^ the same matrix 
correlated to the second suffix, the outer product E^F V is a 16-rowed matrix.J 
An example of a nonad is 

iVi, iE 3l F l} iE l2 F l9 iEtF 29 iE,F 29 iE^F 29 F 39 JF 4 , F 5 . (3-57) 

It is constructed by means of a pair of conjugate triads of E matrices (see 
(3-82)). 

3-6. Determinant of an A 7 -number. 

It is useful to have before us the explicit expression for the matrix which 
represents a general JE7-number T = S ^ E^ with some standard identification 
of the matrices E^ . The following is the matrix representing T, when the 
matrices E l9 E 2y JE? 3 , E^ E^ are taken to be i8 u , iDa, iS y D y , 8^, 8 y Do as 
in (3-27). We write r u for ft u . 

r* r* 



* 6 + T 3 , 
-T 2 -T 14 , T 62 + e 23 -f 6 ~Tg, T 43 + t^ - T 16 - t 3l , 

+*5 ~ T 3J T 16~ T 35"^ T 2 ~ T 14> T 1 ~ T 42 + ^21 "" *4 > 
T 52"~'23"~^5 + T 3> T 43~^45 ~ T 15^'31> T l ~ T 42 ~ ^21 + ^4 T 16 "" T 85 "" T + T 14* 

(3-61) 

The columns correspond to the first suffix and the rows to the second suffix 
of V 

The determinant formed by (3-61) can be evaluated. It is found to be 
detr = S^2E^^ + 8S ^^ TU ^ 8S ^^^^ 6< (3 . 62) 

Here the first two terms on the right are written in single-suffix notation, 

t Journ. Land. Math. Soc. 7, 93, 272 (1932). 

J Matrices of the form S^ are treated fully in Chapter x. 



3-6] The, Resolution of Matrices 43 

and the last two terms in double-suffix notation. In the second term the 
sign is positive iiE^,E v anticommute and negative if they commute; in the 
last term the sign is positive if /A, v, a, T, X, p is an even permutation of the 
suffixes 0, 1, 2, 3, 4, 5 and negative for an odd permutation. It is understood 
that each component is written in one way only in the double-suffix nota- 
tion; e.g. 2 i may also be written as 12 , but it is not to be included a second 
time in the summation on account of the two ways of denoting it. 

The determinant of a mixed tensor is unaltered by tensor transformations. 
Thus (3*62) will be invariant when the special matrix frame used in (3*61) is 
changed to any other frame E ' 

We define the determinant of an E -number to be the function (3-62) of its 
sixteen coefficients. With this definition the determinant of an JSJ-number 
is the same as the determinant of any matrix representation of it; and 
properties of fourfold matrices which involve their determinants can be 
extended to general symbolic JB-numbers. 

It is well known that the condition that a matrix T shall be singular is 

detT^O. ' (3-63) 

Also, by a well-known theorem, for any two matrices S and T 

det(ST) = detSxdetT. (3-64) 

We see by inspection that when E^ = S a D b 

det Ep = det 8 a x det D b = 1 ; (3-65) 

and, since the determinant is invariant for tensor transformations, this 
holds for the matrices E^ of any complete orthogonal set. (The same result 
is also found directly from (3-62).) 

By (3-64) and (3-65) det (E^ T) = det T. (3-66) 

And by (3-62) det (a + bE^) = (a 2 + 6 2 ) 2 (/x ^ 16). 

Hence 

det(cos0 + ^sin0) = l (^16). (3-671) 

We generally write cos + E^ sin 6 = e** e (see 4- 1) ; hence by (3-64) 

det (Te E e ) = det T (^ 16). (3-672) 

We shall later consider transformations of the form T-+Te E n 9 n f 
T-*el B 9 *Te* E n e n, etc. By (3-672) any number of these transformations 
leaves det T invariant, if the algebraic transformation /LC= 16 is excluded. 
Further, if p, = 16 and 16 is real, the transformation does not alter the 
modulus IdetTj. A transformation which leaves |detjf| unaltered is 
called a unitary transformation. Of the 32 possible transformations e E i* 9 
(counting real and imaginary 0^ as different transformations) the only one 
which is not unitary is that given by imaginary 16 . 

These results apply to general ^-symbols as well as to matrices, the 
determinant being defined bv (3-62). 



44 Wave-tensor Calculus [3-7 

3-7. Eigensymbols and Eigenvalues . 

If X is any symbol, and is a symbol (not zero) such that 

X = oc<, (3-71) 

where a is an algebraic number, < is said to be an eigensymbol of X and a an 
eigenvalue of X . We collect here some of the most important properties of 
eigensymbols. The results are given for final eigensymbols; but there are 
in all cases corresponding theorems for initial eigensymbols defined by 
(f) X == a^. 

Iff is a polynomial function, repeated application of (3-71) gives 

/(*).*-/().*. (3-715) 

(a) If the symbol X satisfies a polynomial equation /(X) = 0, the only 
possible eigenvalues of X are roots off (a) = 0. For we have / (X ) . </> = 0, and 
therefore /(a) .< = 0. Then, since ^ is not zero, /(a) = 0. 

In particular if X 2 is algebraic and equal to ra 2 , X has only two possible 
eigenvalues m. The eigenvalues of the E^ are i. 

(b) A symbol which has an eigenvalue has no reciprocal. For if X<f> = 
and X~ 1 X = 1, we have = X" 1 (X</>) = </>. But an eigensymbol, by definition, 
is not zero. 

Hence if an ^-number or matrix T has a zero eigenvalue, it is singular, 
anddet!T = 0. 

(c) If T has an eigenvalue A, T - A has an eigenvalue 0, so that 

det(T-A) = 0. (3-72) 

Accordingly the eigenvalues A of an ^-number or matrix are the roots of 
equation (3-72), which is called the characteristic equation. For an -B-number 
or fourfold matrix the characteristic equation is of the fourth degree in A, 
and may be written 

/(A)-(A-A 1 )(A-A 2 )(A~A 3 )(A-A 4 ) = 0. (3-73) 

It is known that a matrix satisfies its own characteristic equation (Hamilton- 
Cayley theorem), so that we have 

/(r) = (r~A 1 )(r-A 2 )(T-A 3 )(T-A 4 ) = 0. (3-74) 

Equation (3-74) may be regarded as the converse of the result (a). 

The polynomial equation of lowest degree satisfied by a symbol T is 
called the minimum equation. Since every eigenvalue must be a root of the 
minimum equation (by (a)) and every root of the characteristic equation 
is an eigenvalue, the minimum equation can only differ from the character- 
istic equation if the latter has repeated roots. 

(d) To find an eigensymbol of T corresponding to one of its eigenvalues, 
say A 19 we proceed as follows. Let m (T) = be the minimum equation, and 
let sr(T)=m(T)/(J T -A 1 ). Since (I 1 -^) is a factor of m(T), g(T) is a poly- 



3-7] 



The, Resolution of Matrices 



45 



nomial in T\ it cannot vanish, because g(T) = Q would be a polynomial 
equation of degree lower than the minimum equation. Let <l> 
where x is any symbol. Then 



Hence T<j> = \<j>\ so that <f> is the required eigensymbol. 

(e) Mutually commuting matrices S, T, 17, ... have a common eigen- 
symbol, and in particular a common eigenvector. 

We form polynomials ft (5), g*(T) y g^(U) for S 9 T, U as in (d), and take 
<!> = </i(S)9*(T)<J3(U)x- Since ft (S), jr f (5T), <7 3 (#) commute, it follows as 
in (d) that is an eigensymbol of 8 and T and U. Also ft (S), ft (T), g^(U) 
are matrices, and their product is a matrix; hence if x is a vector, (f> will be 
a vector. 

(/) If X and F have a common eigensymbol <, we have (X Y - YX) (f> = 0, 
so that JC 7 7X is either singular or zero. 

(g) IfEp and E v commute, and <f> is an eigensymbol of aE^ + bE v , where 
a 2 7* 6 2 , then <f> is an eigensymbol of E^ and ^ . 

For if (aEp + &#) ^ = a^, 

we have, on multiplying by aEp bE v , 



so that < is also an eigensymbol of aE^ - 6^ . Hence is an eigensymbol of 

+ 6A;) (aB^ - 6A 1 ,), i.e. of E^ and ff, . 
(h) If J^, #, ^? mutually commute, and <f> is an eigensymbol of 



where a 2 ^ fc 2 ^ c 2 , then <^ is an eigensymbol of E^ , ^ and E a . 

By (2-36) and (2-343) JS7 a = iE fJL E v . Hence, if c' = ir, the datum is 



We obtain, on multiplication by E^, E v , 



Hence the determinant 



bEp o/E v oc Ei JS 
j. 


7 y + c / )^ = 0. 


LU 

a 


6 


c' 


a 


= 0. 


a 


-c' 


b 


a 




-c' 


a 


a 


-6 




-6 


a 


a 


c' 





This gives the possible eigenvalues a. Then, by eliminating E and E^E V 
from three of the equations, we obtain a result of the form (jE^ -f k) </> = 0. 
Hence if j ^ 0, ^ is an eigensymbol of E^ . 



46 Wave-tensor Calculus [3-7 

The singular case, when j= for every combination of three equations, 
occurs when the four minors of the fourth column of the above determinant 
all vanish. It is easily found that this condition requires that two of the 
quantities a 2 , 6 2 , -c' 2 shall be equal. This is excluded by the enunciation 

(i) If X, Y 9 Z are commuting symbols with eigenvalues x it y^, z ( 
(i=l, 2, 3, ...), the eigenvalues of any rational function f(X, Y, Z) are 
included among the quantities f(x i9 j 

We can write 



f(X, Y, Z)-f( Xi , y f , z k )={f(X, Y, Z)-f(x { , Y, Z)} 

+ {f(x t , Y,Z)-f( Xi , y } , Z)}+{f( Xi , y it )-/(* y,, z k )} 

z k ), (3-75) 



since the first bracket vanishes when X = x i , the second when Y=y it the 
third when Z=z k . Now form the product 

U{f(X,Y,Z)-f(x t ,y i ,z k )} 

i,3,k 

for all combinations of values of i, j, k. By substituting (3-75) in it, we 
express it as the sum of a number of terms containing products 

(X-x 1 Y(X-x 2 )>...(Y-y 1 )>(Y-y 2 )...(Z-z 1 )(Z-z 2 r.... 

Every term will contain a complete set of eigenvalues of at least one of the 
symbols X, Y, Z. For, if not, let there be a term which does not contain the 
factors (X x t ), (Y-y m ), (Zz tl ). But one of the factors is 



so that either (X x t ) or ( Y y m ) or (Z z n ) must appear in every term. 

The minimum equation for X is m (X) = ll t (X - xj = 0. Since every term 
contains m (X) or m ( Y) or m (Z), every term vanishes; and we have 

n {f(X, r f Z)-/(s <f y, f %)} = <>. (3-76) 

$t j * 

This is a polynomial equation satisfied by the symbol f(X, Y, Z), and its 
roots f(x i9 y i9 z k ) accordingly include all possible eigenvalues of the symbol. 
Not every root will be an eigenvalue; for example, if JC, Y, Z are fourfold 
matrices, f(X, Y, Z) will be a fourfold matrix, so that not more than 4 of the 
64 roots of (3'76) can be eigenvalues. 

Of the above results (a), (&), (d), (e), (/), (i) apply to all symbols which 
satisfy a polynomial equation. A common example of a symbol which does 
not satisfy any polynomial equation is djdx. 

t Frobenius's theorem. It holds for any number of commuting symbols; we here take 
three as a sufficient illustration. 



3-8] The Resolution of Matrices 47 

3* 8 . Paul! Matrices . 

A notation for the U-symbols based upon the conjugate triads (2-37) is 
sometimes useful.f Denote the symbols E^, E va9 E afl \ E r \, E^ 9 E pr9 which 
form conjugate triads, by 

A l9 A 29 A 3 i B l9 B 2 ,Bs. 
Then the sixteen E^ can be written as 

A a9 B 09 iA B r , i (cr,T=l,2,3). (3-81) 

Each A is the product of the other two A\ and each B is the product of 
the other two J3's. The rules of commutation are: an A anticommutes with 
an A, and a B with a B\ an A commutes with a B. 

There is one pair of conjugate triads in which all six matrices are real, viz. 

^23 > ^31 > ^12 ^45 > ^5 ^4> (3' 82) 

J5? 4 , JS7 5 being the real matrices of the pentad. The other ten matrices of the sot 
are imaginary. With this identification, the real or imaginary character of 
the matrices in (3-81) is explicitly indicated by the absence or presence of i. 
This method of constructing a complete set can be exhibited in another 
way. We apply the treatment of 2*2 to two symbols instead of four. Let 
A l9 A 2 be any two symbols which satisfy 

V- 1 ' A^^-ArAp. (3-83) 

Then, if A 3 = A 1 A 29 A 3 is an additional symbol satisfying (3-83). The 
s y mbols A 19 A, A, i (3-84) 

form a "minor complete set". Calling any linear function of them an A- 
number, the operations of addition, subtraction and multiplication applied 
to A -numbers always yield A -numbers. 

A minor complete set can be represented by twofold matrices. Let 

It-*' 0, 2 = i, ^0 -1 

-i i 10 v ' 

These satisfy C/--1, ,,-&- -W M , ( 3 ' 86 ) 

where /*, v, X are in cyclic order. Thus ^ , 2 , 3 , i is a particular representation 
of A l9 A 29 A 39 i. The matrices (3-85) are called Pauli matricea.% 

We can show, in the same way as for the E -symbols, that with this 
identification every .4 -number is represented by a twofold matrix and every 
twofold matrix can be expressed as an ^4 -number. We find also a transforma- 
tion theory for minor complete sets analogous to 2-7, viz. 



where 



f This was pointed out by G. Lemaitre. 

% Most writers employ the matrices tj u whose squares are + 1. 



48 Wave-tensor Calculus [3-8 

In Case (6), in which the ,/ are new symbols commuting with the i^, we 
have a = i, so that the transformation operator is 

P-^-K-l + W^ + kk' + W.'). (3-87) 

To reproduce the sixteen E^ we require two minor complete sets A and B. 
These may be represented by two sets of Pauli matrices ^, 0^, provided 
that the products ^O v are taken as outer products. The outer product of 
two twofold matrices is a fourfold matrix. We are thus led back to the 
ordinary representation of the ^-symbols by fourfold matrices. 

3-9. Left-handed Frames. 

The coefficients t^ of an ^-number are in general complex algebraic numbers, 
sa Y T /i + **ii For * we have a choice of two algebraic square roots of 1, 
which we shall call ij, t a . We may regard the symbols (l,i) as constituting 
an algebraic frame; we have then to distinguish two possible algebraic 
frames (1, ij, (1, i 2 ) either of which can be combined with a given symbolic 
frame E^ . 

We have hitherto used Z? 16 and i indiscriminately; but it is now desirable 
to define the structure of an orthogonal symbolic frame unambiguously by 
eliminating i in the fundamental equations. We therefore replace (2-22) 
and (2-343) by (3-91) 



where p, i/, cr, r, A, p is an even permutation of 0, 1, 2, 3, 4, 5. 

An .E-nimiber T = (r^ + iv^) E^ will involve an algebraic square root of 
- 1 denoted by E 19 which occurs in the symbolic frame, and also an algebraic 
square root of - 1 denoted by i which occurs in the coefficients. The complete 
reference frame for ^-numbers thus consists of 

(a) A symbolic frame E^ , 

(6) An algebraic frame (l,t) for the coefficients. 

jB 16 and i may or may not be the same root of 1. Absolutely it is meaningless 
to inquire whether they are the same root. But if we (arbitrarily) regard 
them as the same in one complete reference frame, we can define other 
complete reference frames in which they are opposite roots. 

A complete reference frame in which E l9 = i will be called right-handed, 
and a frame in which E u = - i will be called left-handed. 

Since E le is invariant for tensor transformations, there is no transforma- 
tion of the type F^qE^q' between right- and left-handed frames.f Their 
relation is like that of right- and left-handed systems of rectangular co- 
ordinates, which cannot be changed into one another by rotation. 

We usually treat the algebraic frame as unalterable so that a right- 
handed frame E^ and a left-handed frame F^ are distinguished by F^ = - E^ . 

t The existence of two kinds of complete sets not transformable into one another by 
relativity transformation was, I think, first recognised by 8. R. Milner. 



3-9] The Resolution of Matrices 49 

The relation of the other matrices can have various forms; we give the three 
most important. If the frames are constructed from the same tetrad 

we have by (3-91) lf 2 ' 3> 43= lf 2l 3> 4f 



(3-92) 

the other ten matrices being the same. Similarly if they have the tetrad 
E u , E 25 , E 3S , E^ in common, 

^18> ^l ^2> ^3 -^4> ^5 ""^18 "~-^l> "~^2> ~~^3 ~"^4 "" ^5 > (3'93) 

the other ten matrices being the same. Another form, obtained by giving 
alternative signs to i in (3-81) is 

F r = E r , *l=-0 if (3-94) 

where E r denotes the real matrices 2? 23 , E$ l9 B 12 , E^ E 6 , E^, and A\ the 
imaginary matrices. In this last case we change from a right- to a left- 
handed set by writing i for i in the elements of the matrices. 

We shall find later that the distinction between positive and negative 
electric charges corresponds to the distinction between right- and left- 
handed frames; so that a positive charge cannot be changed into a negative 
charge by a relativity transformation of frame. 



CHAPTER IV 

SPACE VECTORS 
4-1. Rotations. 

When an exponential contains non-algebraic symbols, it is understood to be 
denned by the exponential series. Thus 



since JSL 2 = 1. Hence 

eV=rcos0 + J^sm0. (4-11) 

In fact, so long as no opportunity for exhibiting non-commutative pro- 
perties arises, E^ is indistinguishable from i. The reciprocal of e E G is e~ E ^. 
The ordinary factorisation of an exponential e a +0 = e a .e holds only so 
long as a and ]8 commute. For example, 

e K 1 e+E S3 <t = e E 1 9 me E^ 9 e *i0+^4 96 e*i*.eM. (4-115) 

By (4-11) 

ePvflEy = (cos 6 + Ep sin 6) E v 

= E v (cos + Ep sin 0) if EH , E v commute 
= E v (cos Q Ep sin 0) if E^ , J57 y anticommute. 
Hence eP e E v = j0,eV if E^ , JS7 V commute ] 

= E v e- E 6 if JS^ , E v anticommute/ " ( ' ' 

We now consider the relativity transformations of the JS?-numbers 
describing a physical system. By (2-94) the change t^t^ of the coefficients, 
due to a rotation of the physical system relative to a fixed frame E^, is 
given by 

Let } 

which satisfies qq' = 1. Then 



J^e-*^, (4-14) 

by (4-12), where S a denotes summation of the eight terms which commute 
with E 12 , and 2^ summation of the eight terms which anticommute. As an 
example of terms anticommuting with E 12 and therefore included in S^, 
we take 1^ + 1^. We have 

a ) cos - (t l E l + t 2 E 2 ) JS? 12 sin 
+ t 2 E 2 ) cos fl - ( - ^ JS? 2 + < 2 EJ sin fl 



4-1] Space Vectors 51 

By 2-4 (6) we may equate coefficients of the same matrix on both sides of 

(4*14) Hence 

tj = ft cos 0-J 2 sinff, ^'^sintf + facosfl. (4-15) 



That is to say, the relativity transformation q = e*** rotates (^ , t 2 ) through 
an angle 0. 

Examining the other terms in S^ we find that three other pairs of com- 
ponents are rotated in the same way, viz. < 13 , t^\ t u * ^4* ^is > ^25 The remaining 
components are unchanged. 

Considering the more general relativity transformation g=e^^, and 
taking /z = 1, 2, ... 15, we obtain 15 independent rotations of this type, each 
rotating four pairs of components. We may, if we like, add the E 19 rotation, 
viz. q = e& e , q' = e~t i6 , but this does not alter T. 

Two terms which mutually rotate necessarily anticommute with each 
other as well as with the transformation matrix. We can always find a 
transformation matrix which will rotate two anticommuting terms E^ vt 
Ep a , v * z - their product E va . There is no corresponding mutual rotation of 
two commuting terms. Suppose, for example, that we try to rotate E^ and 
^23*23- Since J? 23 = iE^E lt we shall require the transformation q=e* K *& ', 
but E l and J? 23 commute with J0 45 , and therefore come under S a . Accordingly 
^ , J 23 are unchanged by the transformation. 

Pairs of components which can rotate with one another will be called 
perpendicular] pairs which cannot rotate will be called antiperpendicular. 
We have the rule: 

Matrices commute: components antiperpendicular; 
Matrices anticommute : components perpendicular. 

The term orthogonal will be understood to include antiperpendicularity as 
well as perpendicularity. 

If the components ^ are represented as coordinates in a space of 16 
dimensions, the space contains certain planes in which rotation is possible 
and certain planes in which it is forbidden. Actually the 120 coordinate 
planes consist of 60 planes of rotation and 60 forbidden planes. More 
precisely we should say that rotation in a forbidden plane is a non-relativistic 
change; although depicted graphically as a rotation, it is an intrinsic 
deformation of the physical system described by t^ . 

Rotation in a forbidden plane will be called an antiperpendicular rotation. 
It is produced by the (non-relativistic) transformation 

(4-16) 



or t 

Taking g=e* B * as before, we have instead of (4-14) 



52 Wave-tensor Calculus 

Hence pairs of terms which are not rotated by the relativity rotation are 
rotated by the antiperpendicular rotation, and vice versa. As an example of 
terms commuting with E 12 and therefore included in S a , we take 



= (t 3 E 3 + 45^45) cos + (^3^45 + ^45^3) sin 9 
= ( 3 cos + i 45 sin 6) E 3 - i ( - 3 sin + # 46 cos 

Hence t 3 ' = J 3 cos 4- i 45 sin 0, i 46 ' = 1$ sin -f i 45 cos 0. (4-17) 

The other pairs rotated by the same transformation are 

^4> *^35 ^5 ^34 ^12 > ^16' 

If alternatively we treat (4-16) as a transformation of the frame, so that 
T ' = S W> where */-, (4-18) 

the new frame S^ does not satisfy the conditions (2-23) for a complete 
orthogonal set. Its structure is intrinsically different from that of the 
standard frame E^ and the physical structure built by t^ in such a frame is 
therefore not equivalent to the structure built by the same t^ in an orthogonal 
frame. It is to be noticed that this argument that antiperpendicular rotations 
are non-relativistic changes, does not introduce the question whether (4- 16) 
is a tensor transformation. We shall see later (Chapter vn) that in some cases 
(4*16) is a rather simple tensor transformation, though it is not that of a 
mixed tensor. I think that confusion of thought has often been caused by 
failure to recognise that, although tensor calculus is an almost indispensable 
tool in relativity theory, it does not in itself imply any relativistic hypo- 
thesis. 

4- 2 . Alternative Treatment . 

The rotations of the components t^ can also be found directly from (3-35) 
or (3-37). Consider first a factorisable matrix J = 0x* = s j ft J^. By (1-461) 
and (1-462) the relativity transformation g = eHA gives 

0' = eH**M ^, x *' = x *e-k*A. 
Hence, by (3-37), 

j v ' = - lx* r vV = ~ i x *e-**A^e*VM ^. 
If JE^, E v anticommute, this becomes, by (4-12), 



where J a == JS^. Hence, by (3-37), 

3* =^ cos 9 4- j a sin 0. (4-21) 

If Ep , E v commute, the result is j v ' =j y . The components of a general matrix 
T are transformed according to the same formula, because the general 
matrix can be expressed as the sum of a number of factorisable matrices. 



4-3] Space Vectors 53 

It is tempting to combine these elementary rotations into a "general 
rotation" g = e* 2J5 A; but we have seen in (4-115) that non-commuting 
exponentials cannot be compounded in this way. This prohibition merely 
reflects the non-commutability of rotations in ordinary Euclidean geometry , 
and is not a peculiarity of the -spaee. The objection, however, does not 
apply to a combination of infinitesimal rotations. The most general in- 
finitesimal rotation, corresponding to a general infinitesimal matrix dfc), is 

g = e* d = eteV^, (4-22) 

the squares and products of dB^ being neglected, so that the question of non- 
commutation does not arise. Equivalently (4*22) may be written 

q = 1 + $d = 1 + iSB^ . (4-23) 

When the ^ are represented as coordinates in a 16-dimensional space, 
each elementary relativity rotation g = e*M/* appears as a rotation through 
an angle 0^ occurring simultaneously in four different planes. The 15 values 
of /z (excluding ft= 16) accordingly give 60 planes of rotation; and there 
remain 60 forbidden planes. It is evident that the geometry of this 16-space 
is different from that of any type of space ordinarily studied. The novel 
features are (1) the occurrence of antiperpendicular pairs of axes, and (2) 
the locking together of four rotations. We shall now consider how to repre- 
sent the components t^ in a more familiar type of space. 

4*3. Five -dimensional Euclidean Space. 

A pentad provides five mutually perpendicular components t l9 1%, t^, J 4 , J 6 . 
Any pair of these can be rotated. Moreover, the rotation matrix E 12 com- 
mutes with E 9 , 7? 4 , E 5 , so that the mutual rotation of ^ and t z leaves 3 , 4 , J 5 
unaltered. It is therefore a simple Euclidean rotation so far as these five 
components are concerned. 

If then we represent (t l9 2 > ^a> ** W as coordinates (or as components of a 
vector) in five-dimensional space, this space has the ordinary relativistic 
properties of Euclidean space, namely that simple rotation in any of the 
ten coordinate planes is a relativistic transformation. Accordingly the 
intrinsic properties of a physical system are not affected by changing its 
orientation in this space. 

Thus by limiting ourselves to a sub-space of five dimensions we encounter 
neither of the complications of the geometry of 16-space. The domain of 
(^u 2 a , J 3 , 4 , < 5 ) has the properties which we attribute to ordinary space; and 
(leaving aside the fifth dimension for the present) we may identify physical 
space-time with a continuum constructed in this way. 

This provides a linkage between wave tensors and space tensors (1*2). 
A space vector is the pentadic part of a mixed wave tensor . Ordinarily we regard 
the components of a space vector as an array (t l9 t 29 t^ 9 t 4t9 6 ), whereas in a 



54 Wave-tensor Calculus [4-3 

wave tensor they are strung together with symbolic coefficients as a linear 
expression /^ + t z E 2 -h t$ E z + 4 1? 4 + J 6 J? 5 ; but the latter mode of representing 
space vectors has long been recognised as permissible, e.g. in quaternion 
notation, so that the difference of form need not be stressed. 

This is not a hypothetical identification. In 1-2 we left the basic wave 
vector undefined, and we are therefore free to define it at this stage. We 
now define the relation of wave tensors to the ordinary space vectors of 
physics to be such as is expressed by this identification. Henceforth our 
calculus embraces both wave tensors and space tensors. 

The question remains, What is the significance of the eleven remaining 
components of the wave tensor? When the 5-vector undergoes a relativity 
rotation, e.g. in the plane t t t 2 , rotations also occur between 13 , 23 ; 14 , t^\ 
*i5 ^25 The ordinary tensor calculus provides for these locked transformations ; 
a transformation consequent on a transformation of the basal space vector is 
made automatic by assigning the appropriate space tensor character to the 
quantities concerned in it. We shall prove in 4-5 that the following is the 
required specification: 

(a) 19 2 > *3> *4> 5 is a space vector, 

(6) 12 , 13 , ... 45 is a 10-vector or antisymmetrical space tensor of the 
second rank (analogous to a 6-vector in four dimensions), 

(c) 16 is an invariant. 

The statement that (a) is a vector and (b) an antisymmetrical tensor of the 
second rank secures that the transformations of (b) are locked to those of 
(a) in such a way that the rotations of pairs of terms occur in groups of four 
as required. 

We shall call the group of space tensors (a), (6), (c) a complete space vector, 
or (if no ambiguity is likely to arise) simply a space vector. We have therefore 
the simple relation 

Mixed wave tensor = Complete space vector. (4-31) 

In dealing with space vectors we recognise only ten relativity rotations, 
viz. the ten rotations of five-dimensional space. The wave tensor had 15 
relativity rotations (or 16, if we count the algebraic rotation). The five 
extra rotations intermingle the two space tensors (a) and (6); for example, 
the E 2 rotation rotates the component ^ of (a) with the component t l2 of (b). 
Ordinarily (a) and (6) will be recognised in physics under different names, e.g. 
velocity and spin, and a transformation which does not keep them distinct 
could only be pictured at the cost of abandoning the usual representation 
in space and time. We rather miss the point of the interpretation of mixed 
wave tensors as space vectors, if we go on to attribute to the space vectors 
transformation properties outside those which the name ordinarily suggests. 
There is no advantage in introducing space vectors so generalised that we 



4-4] Space Vectors 55 

can no longer " see " them in space; for if we are not going to use the space 
picture, we have to fall back on their analytical description as mixed wave 
tensors, and it is more appropriate to refer to them by that name. We may 
therefore agree that the name space vector implies that only the ordinary 
relativity rotations of the space are under consideration; and that if it is 
desired to include the transformations which transcend space-time repre- 
sentation the proper designation is wave tensor. 

This question of nomenclature arises because in 7-6 we shall define an 
important entity which behaves as a mixed wave tensor for the ten rotations 
of 5-space, but not for the other rotations . Thus it may be properly described 
as a space vector, but not as a mixed wave tensor. Except in this connection 
it is not necessary to emphasise the distinction, and I shall generally use 
the two names as equivalent. 

4*4. Four -dimensional Spherical Space. 

We have now to consider why the actual world is four-dimensional, although 
the analytical theory (which we have reason to think is appropriate to the 
physical world) seems to provide for five dimensions. The answer is not 
difficult to find. Space-time is four-dimensional, but it is not flat (Euclidean) ; 
and if we restrict ourselves to Euclidean geometry, we require at least one 
more dimension to represent its curvature. 

Our actual space-time with its irregular curvature, due to local gravi- 
tational fields, requires a ten-dimensional Euclidean space for its repre- 
sentation; but we do not arrive at this complexity until we provide for a 
more varied content of the universe than can be represented by a simple 
wave tensor. Any problem which is mathematically simple is necessarily 
highly idealised; and the simple wave tensor with which we commence our 
study can carry us only a little way towards actuality. We have found 
that it is capable of rotation, without intrinsic change, in ail planes in 
five dimensions. This leads immediately to the conception of an entity 
distributed over a continuum which is a hypersphere (four-dimensional 
manifold) in a Euclidean space of five dimensions. We must start with this 
simplified space-time, and watch the more complex characteristics of actual 
space-time grow out of it as the theory develops. 

We shall first show that it is the hypersphere (and not the five-dimen- 
sional space in which it is represented) which constitutes the physical 
continuum. This is because the hypersphere is a locus of equivalent 
points, whereas the points in five-dimensions are not generally "equivalent " 
points. 

Let the coordinates of a point P be (^ , t 2 , J 3 , $ 4 , 5 ). Any of the ten rotations 
in five dimensions will (if it displaces P) carry P to a new position P 1 on 
the same hvneranhftTe ahniit tha oricrin. Than t.hft -nninfja P P' nr 



56 Wave-tensor Calculus [4-4 

structed according to the same specification in different but equivalent 
frames, and are therefore equivalent points (2-9). None of the relativity 
transformations provides any connection between points which are not on the 
same hypersphere. If we consider a point P" such that PP" is normal to the 
hypersphere, P and P" are not equivalent points. The transformation 
(0, 0, 0, 0, J 6 ) -> (0, 0, 0, 0, t$) is not a relativity transformation; it is in fact 
an antiperpendicular rotation with matrix A 7 16 . 

Thus displacement normal to the hypersphere involves, either a different 
construction, or the same construction in a frame which differs intrinsically 
from the original frame. Taking the latter view, the difference between the 
two frames is evidently a difference of scale-constant. Our five-dimensional 
picture accordingly represents change of scale-constant or gauge by dis- 
placement in a fifth coordinate, normal to the four ordinary coordinates 
used to represent position. A similar graphical representation of change of 
gauge is used in "Projective Relativity". 

It is the essence of the elementary conception of space and time that all 
points of it are equivalent. A particle is not intrinsically different because it 
is at a different point of space or because it is contemplated at a different 
time. It is true that in later developments regions of space and time are 
distinguished from one another by varying curvature; so that the space-time 
background is no longer a "blank sheet". But we have to trace the origin 
of these distinctions, and must begin with the blank sheet. This is provided 
by the hyperspherical continuum of equivalent points; the curvature is 
uniform, and every part of the continuum is precisely similar to every other 
part, as the conception of " equivalence" implies. 

Thus although the transformation theory introduces the conception of 
five dimensions, it is clear from the start that there is an absolute distinction 
between the displacements lying in the four-dimensional hypersphere and 
displacements in the fifth dimension normal to the hypersphere. Whereas 
the former changes of a system are of the type which we conceive as dis- 
placement in physical space and time, the latter are only " displacements " 
in the sense in which we regard any change of quality of a system (scale, 
temperature, entropy, etc.) as a displacement of the representative point 
in a graph of that quality. 

Let us now confine attention to a region of the hypersphere small enough 
to be treated as flat, and let t b be the coordinate normal to this region, the 
coordinates in space-time being (t l9 t 2 ,t 3 , J 4 ). We now exclude the rotations 
^15 > ^25 ^35 > ^45 > because they would carry us away from the small region 
considered and mix the scale-coordinate 5 with the coordinates recognised 
in our ordinary outlook. There remain the relativity transformations 
corresponding to the matrices 



4-5] Space Vectors 57 

These constitute the transformations admitted in special relativity theory, 
viz. three rotations in space and three Lorentz transformations. We now 
consider how the general mixed wave tensor will appear from this outlook 
the ordinary outlook on which current nomenclature is chiefly based. 
The mixed wave tensor is found to break up into 

(a) a vector t l9 t 2 , J 3 , ? 4 , ^ 

(6) another (adjoint) vector 15 , J 25 , J 35 , J 4B , | (4-42) 

(c) a 6-vector * 23 , * 

(d) two invariants 

The proof that (a) and (6) undergo the same transformations for any of the 
rotations (4*41), and are therefore "tensors of the same kind", is obvious. 
The proof that (c) is a 6-vector is given in 4-5. 

Thus in four dimensions the complete space vector is composed of two 
vectors, a 6-vector, and two invariants. These remain distinct in any of the 
six internal rotations of the 4-space which transform the 4-space into 
itself. Naturally the use of the four-dimensional picture presupposes that 
we are not interested in the transformations which cannot be represented 
in the 4-space; if we have occasion to refer to them we must revert to one 
of the other modes of description. 

If we are right in our belief that all physical phenomena are analysable 
into ultimate elements described by wave vectors and their combinations, 
it follows that an ordinary space vector cannot occur alone; it is part of a 
group of allied space tensors (4-42). The other members of the group may, 
of course, have zero value, but that is not the same as being non-existent. 
This is one of the ways in which the new outlook enriches the earlier theory. 

4-5. Proof that (J 23 , / 31 , / 12 , 14 , J 24 , J 34 ) is a 6-vector. 

If Xp , y^ (p = 1, 2, 3, 4) are ordinary space vectors X, Y, their vector product 
consists of six quantities (x^yv ^y^). Then any set of six quantities which 
transform according to the same law as (x^ x^y^) is called a 6-vector. 
In matrix form the vectors are 
X = E 



Their matrix product X Y is found by direct multiplication (using E^ = - 1 , 
EpE,- -tf,ig to be XY = - 

where (xy) is the scalar product, and 



to be - (4-51) 



). (4-52) 

Apply a transformation q representing a rotation or Lorentz transforma- 
tion in four dimensions. Then (xy) is invariant, and 



58 Wave-tensor Calculus 

Hence by (4-51) S' = X'Y' + (xy)=qXYq' + (xy) 



(4-53) 

Let # -# 2 3*23 + ^31 '31+- +^34*34^ ( 4 ' 54 ) 

Then, since the constituents (a), (6), (c), (d) of the complete space vector T 
transform separately, we have ^, _ jj , (4-55) 

By (4-53) and (4-55) U and S obey the same transformation law for rotations 
and Lorentz transformations in four dimensions. It follows that their 
components t^ v and (x^y v - x v yj obey the same transformation law. Hence 

Vv 0*' v== lj 2 > 3 > 4 ) is a 6 - vector - 

We can show similarly that t^ v (ft, v = 1, 2, 3, 4, 5) is a 10-vector in 5-space. 

It will be seen that the matrix product XT, used in the present calculus, 
is not the same as the ordinary vector product X x Y, but is X x Y - (xy). 
When two vectors are perpendicular their scalar product (xy) vanishes, and 
the matrix product is then the vector product. 

The product of two complete space vectors is a complete space vector. 
For the product W UV transforms according to the law 

W = U'V'=: qUq'qVq' = qUVq' = qWq'. (4-56) 

But, as we have seen above, the matrix product here employed is a type of 
combination which has no exact counterpart in the ordinary theory of 
vectors. 

4-6. Volume Elements. 

The expressions E^dx^ E 2 dx 2 , E^dx 39 E^dx^ are the wave tensor notation 
for four space vectors (displacements) along the four rectangular axes in 
space-time. Their product constitutes a volume element dV I234k . Writing 
dr = d 



by (2-22). Since the jEJ's anticommute, dV^^ is antisymmetrical in its four 
suffixes as in the ordinary tensor calculus. f The factor iE 5 corresponds to 
V - g. We are restricted to rectangular coordinates (one of them time-like), 
and therefore g is always 1; but we deviate from the usual theory which 
assumes that the radical indicates the algebraic square root. We take 
instead a matrix square root iE 5 . 

Consequently, notwithstanding that we use rectangular coordinates, the 
distinction between vectors and vector densities does not wholly disappear. 
In the ordinary_calculus we have, corresponding to a vector A , a vector 
density A^V g. Here V g = iE b , so that corresponding to a vector 



we have a vector density 
i(E 
t Mathematical Theory of Relativity, 49. 



4-6] Space Vectors 59 

Of the two adjoint vectors in (4-42) one represents a vector and the other a 
vector density. We shall show later ( 5-8) how it is possible to decide which 
is which. 

More generally we define the volume element contained by four space 
vectors ePa?^, &*x^ A^x^ d*Xp, which do not coincide with the axes, to be 
the permanent 



^ 



1234 = 



E 2 d l x 2 , 



(4-62) 



E 2 d 2 x 29 
E 2 d*x 2 , 
E 2 d*x 29 

A permanent is expanded like a determinant except that all the terms are 
given positive sign. The factors are arranged in the order of the rows from 
which they are taken; since the rows correspond to the four vectors, this is 
the natural order of the factors when the vectors are multiplied in a given 
order. The anticommutation of the four J5?'s provides an alternation of sign 
which converts the permanent into a determinant; and (4-62) reduces to 

dF 1234 = E l E 2 E 3 E^ dot (dXp) = iE & det (dxp). 

In our later developments three-dimensional vector densities are more 
important than the foregoing four-dimensional densities. The volume- 
element of three-dimensional space, contained by vectors E i dx 1> E 2 dx 29 
E 3 dx 3 along the coordinate axes, is 

^E 2 E^dw = lEudw, (4-63) 



where dw = dx^dx^dx^ . 
If T is any space vector 

TdW l2 z = iTEudw = Sdw, (4-64) 

where S = iTE^. (4-65) 



Then 8 is the three-dimensional vector density which corresponds to the 
vector T. We shall later meet with S in another connection, in which it is 
called the strain vector associated with the space vector T. 

By (4-65) we can find the componehts s^ of 8 in terms of the components 
tp of jP. The following is the complete scheme of relation: 



The components of S associated with real matrices are given in the first line 
and those with imaginary matrices in the second line. It will be seen that 
the latter correspond to the 10-vector part of T. 



60 Wave-tensor Calculus [4-7 

4-7. Wave Functions. 

Consider a wave vector which is a function of a complete space vector T. 
That is to say, to every one of a continuous set of space vectors T there 
corresponds a wave vector which we denote by ^ =/ ( T), or by/ (^ , J 2 , . . . J 16 ) , 
where t l9 t 2 , ... are the components of T. 

It is usual to distinguish certain of the variables t^ as coordinates, the others 
being parameters of the wave function /. Denoting the coordinates by # , 
and the parameters collectively by a, the notation is changed to $ a =/ a (x^). 
Thus the original wave function is treated as a continuous set of wave 
functions \f> a distinguished by parameters a, each of which covers the domain 
of coordinates # . 

In most practical applications the domain of coordinates is space-time; 
and a wave function has the form ^=/(# l5 tf 2 > a? 3 , o? 4 ), or f(X), where 
Z = ^ 1 x 1 + E^x^ + E^x 3 + B^XI. The function then specifies a wave vector 
field in space-time. 

The function /is necessarily double-valued. To show this, let 



I.e a wave- vector field, r, 0, </> being polar coordinates. If we apply a tensor 
transformation jsae**" 06 to the frame, we refer the vector field to a new 
coordinate system. By (4-15) the effect of the transformation on the co- 
ordinate system is that the point whose azimuth was <f> in the old system 
has an azimuth ^ -f a in the new system. The new coordinates are therefore 

r' = r, 0' = 0, ^' = < + oc, # 4 ' = z 4 . (4-72) 

The transformation of ^ is $' = e^w a ^. (4*73) 

Let the vector field in the new coordinate system be 



,s 4 ). (4-74) 

Then, by (4-73), I" (r, 0, <f> + a, xj = e^JP 1 (r, 0, * 4 ). (4-75) 

Now let a = 277. Since e*" 11 " = 1, 

P' (r, 0, ^ + 27T, Xt)=-P (r, 0, ^ ajj, (4-76) 

or in rectangular coordinates 

/' (*i, x*, ZB, ^ 4 )= -70*1, ^2, ^3, aj- ( 4 ' 7 7) 

But, since the axes after rotation through 2?r are the same as they were 
originally, */>=f(x l9 x 2 ,x 3 , xj and 0=/ ; (x l9 # 2 , # 3 , a; 4 ) are both expressions 
for the wave-vector field in the original coordinate system. Thus iff is a 
double-valued function f of rectangular coordinates. 

Since this result is of vital importance, we must try to remove any 
doubt as to its meaning. Let I! be a particular frame of rectangular 
coordinates and time, and let P be a particular point in that frame. 



4-7] Space Vectors 61 

Suppose first that iff is single- valued; so that, referred to the frame and 
at the point P, it has the value ift and no other; in particular 0^ ^r . 
When we vary the frame or the point considered, iff changes; but if after 
such variation we return to the frame S and the point JP, ifr must return 
to iff Q . For example, if we rotate the frame in any plane in space, after 
a rotation 2ir we come back to the frame S again, so that iff is again . 
But if is a wave vector, the law of transformation of wave vectors 
requires that, when the space-time axes are rotated through 2?r, iff shall 
change continuously from ift a to iff Q . Therefore the single-valued if/ which 
we have been considering cannot satisfy the transformation law of a 
wave vector. 

The double-valuedness of / becomes of practical importance when (as 
usual) the vector field is defined over a region which includes all azimuths <f>. 
For then, on following the point P round a circuit back to the initial 
azimuth, we may find ourselves on the opposite branch of / from that on 
which we started. The return may be either to the opposite branch or the 
same branch ; either type of connection satisfies the condition of continuity 
of ift in the region over which it is defined. 

We note accordingly (for future reference) that a symbol iff^ (a = 1, 2, 3, 4), 
defined to be a single-valued function of rectangular coordinates over a 
domain which includes or encircles the origin, cannot be a wave vector. By 
"encircles the origin" we mean that a circle having the origin as centre 
can be drawn in it. 



CHAPTER V 
THE SIMPLE WAVE EQUATION 

5* 1 . Invariant Equations . 

In order that the laws of physics may be independent of the choice of frame 
(among the equivalent orthogonal frames) they must be expressible as 
tensor equations. In wave-tensor calculus, the simplest non-trivial tensor 
equation is of the form ^ i __ Q / 5.11 \ 

where H is a mixed wave tensor and $ a covariant wave vector. Then (5-11) 
is a vector equation Hj*iltp = Q equivalent to four algebraic equations. 

It may be anticipated that the simplest, and presumably the most 
fundamental, laws of physics will have this form. Alternatively, regarding 
(5-11) as a definition rather than a law, it is an appropriate means of intro- 
ducing a wave vector and relating it to the ordinary space vectors of 
physics. For we have seen ( 4-4) that a mixed wave tensor H is constituted 
of space tensors which will presumably be recognised as such in our practical 
observations; but there is no such "projection" of ^ into a space-time 
representation, and its connection with the ordinary space tensors of physics 
can only be expressed indirectly by an equation such as (5-11). 

It is appropriate to introduce simultaneously a contravariant wave 
vector x*, satisfying x *ff = 0. (5-12) 

We may expect that $ and #* will occur symmetrically in physical theory. 

Thus far our argument has been that if ever nature condescends to 
simplicity, equations of the types (5-11) and (5-12) will figure in her scheme. 

Before the birth of wave mechanics the systematised part of physics was 
wholly described by space vectors and tensors. Wave mechanics introduced 
a new kind of entity 0. It was introduced in the way here proposed by a 
"wave equation" in which the coefficients were the ordinary space-tensor 
quantities of physics. The original ifi of Schrodinger did not satisfy the 
relativity requirements of atomic physics; but in 1928 Dirac introduced a 
/r with four components, which satisfied an equation invariant for the six 
relativity rotations of space-time, although the invariance was not of a kind 
contemplated in the usual tensor calculus. Our anticipated fundamental 
equation turns out to be a form of Dirac 's equation.f 

t The form given by Dirac (Quantum Mechanics, 2nd ed., p. 255, equations (9) and (10)) 
is the equivalent strain vector equation, which we shall obtain in (7'73). Dirac further 
postulates, as a "reality condition", that the two wave vectors are conjugate complex 
quantities. Our reality conditions are determined directly from relativity principles in 
Chapter vi, and do not impose this restriction. 



5-2] The Simple Wave Equation 63 

The mixed wave tensor H used in Dirac's equation is limited to five com- 
ponents. One component has dropped out through special choice of axes, 
permissible when as usual the region contemplated is small enough to be 
treated as flat. Apart from this the truncation is significant, because the 
general mixed wave tensor T cannot be reduced to Dirac's special form // 
by any choice of axes. We shall account for this limitation in 5-4. 

We shall derive Dirac's equation according to the principles which we are 
developing in 5-4; but we shall first examine its elementary properties 
looking ahead to see the theory which we are about to meet. 

Dirac's equations are Hi/t = Q, x*JF/ = 0, (6-13) 

where H = E l p 1 + E 2 p 2 + E 3 j) 3 + ff 4 P 4 ~ m (5- 14) 

and E l9 E 2 , J5 3 , E constitute a tetrad. We call H the Jiamiltonian.^ Since 
// is required to be a mixed wave tensor, its components form space tensors 
in accordance with the specification in (4-42), namely (p l9 p 29 ^ 3 , 2> 4 ) is a 
space vector and the quarterspur m is an invariant. 

Conversely if, following Dirac, we construct a hamiltonian // out of a 
"momentum vector " (p l , p 2 , p* , p*) and an invariant mass m by the formula 
(5-14), H will be a mixed wave tensor; and therefore the wave equations 
/fy = 0, x*// = will be tensor equations which continue to be satisfied 
when any of the six relativity transformations of space-time are applied. 

Thus the invariance of Dirac's equation for relativity transformations, 
which was a novel kind of invariance from the point of view of ordinary 
tensor calculus, is an elementary consequence of wave-tensor calculus. 

5-2. Properties of Dirac's Equation. 

The equation H^ = shows that H has an eigenvalue 0, and hence that it is 
singular ( 3-7 (6)). It has a pseudo-reciprocal 

H'^Eipi + Etpz + Ezpz + Etpt + m. (5-21) 

For, multiplying by (5-14), 

##'= -Pi 2 -P2 2 -P* 2 -P* 2 ~ 2 . ( 5 ' 22 ) 

The product contains no non-algebraic terms, and therefore vanishes by 
(2-66). 

For a physically real momentum vector (p l , p 2 , p 3 , jp 4 ), p l , p 2 , Pz are real 
and PI is imaginary. Let ft = ^ ? (6 . 23) 

f In classical theory the hamiltonian is the expression for the energy ( tp 4 ) in terms of 
the momenta p l9 p %9 p 9 and coordinates. If, following the usual relativistio view, time is 
treated on the same footing with the other coordinates, the hamiltonian is correspondingly 
defined as the expression for the proper energy m in terms of p l9 p 29 p 99 p 4 and the co- 
ordinates. This would make the hamiltonian strictly T+m, but we shall use the term without 
regard to an additive constant. 



64 Wave-tensor Calculus [5-2 

so that ft is the real time component, i.e. the energy or mass. The vanishing 
of (5-22) gives rf- A -- A -- A -- A '. (5-24) 

This identifies w with the proper energy (or proper mass) corresponding 
to the momentum vector. Thus m must be real. 



By (5-24) p Q = (mt+pf+pf+pf)*. (5-251) 

Usually p l9 2h> Pz are small compared with m, and we then have the 
"classical " approximation for the energy 

P Q = m + (Pi* + ft 2 +ft 2 )/2w. (5-252) 

The general solution of the wave equations is found as follows. Let ^, o>* 
be arbitrary four-valued quantities. Since HH' = H'H = 0, we have 



Thus ^ = #'<, x* = w*77' (5-26) 

are solutions of (5-13). Inserting row-and-column suffixes, these become 



Since ^ lf <^ 2 , <^ 3 , < 4 are arbitrary coefficients, the general value of a is a 
linear combination of four elementary solutions # al ', /7 a2 ', ^T a3 ', // a4 ', i.e. the 
four rows of the matrix //'. Similarly the four columns of H' are the elemen- 
tary solutions for # Since H' is singular, its determinant vanishes, and 
therefore only three of the rows and three of the columns are linearly 
independent. There are therefore not more than three independent solutions 
in each case. 

For example, use the matrix representation (3-61). The special form (5-21) 
for H' gives 

-i#'= pz-im Pl +p Q ps 

Pi-Po -ft -fin -#3 

-ft ft-iw PI-PQ 

ft Pi+p Q -p 2 -im. (5-28) 

Any row gives a solution for ^ and any column a solution for x*- It is not 
necessary to choose the solutions in a corresponding way.f For example 
we might take 

^ = (ft~im,p 1 +ft, 0,ft), x* = (0, -ft,ft-im,Pi+ft) 
as the pair of wave vectors constituting a solution of the wave equation. 
t Our treatment here differs fundamentally from that of Dirac. See footnote, p. 62; 



5-3] The Simple Wave Equation 65 

5-3. The Stream Vector. 

Let iff, x* be solutions of the equations H$ = 0, #*/? = 0; and let 

X * = J = S^. (5-31) 

Multiply the wave equations by initial x*^ia an( l final ^12 0> respectively; 
we obtain 



Hence, subtracting, j 

so that, by (3-37), j*Pi-PiJ* = 0- (5-321) 

Again, multiplying by initial x*^i5 an d final # 15 0, and subtracting, 



(5-322) 
so that ^'5 = 0. Also multiplying by initial x*^i an d final E^, and adding, 



(5-323) 
The results (5-321), (5-322), (5-323) give 

3\ _ fa _ Jz _ Ji _3& _ V IB / K oo\ 

X lO'uOJ 

Pi P* Ps P* m ' 

The wave equation can therefore be written in the equivalent form 

(EJi + EJ 2 + #J 3 + ^7 4 J 4 - JP M J 16 ) ^r = 0. (5-34) 

We call (Ji,J2,j* 9 Ji) the stream vector. The whole set of sixteen j^ is the 
complete stream vector. We have here proved that the stream vector is equal 
to the momentum vector except for a numerical factor. Multiplying the 
complete stream vector by the same factor we obtain the complete momentum 
vector. 

We regard this correspondence of the stream vector and momentum vector 
as a coalescence which occurs in the peculiarly simple system here studied. 
In more general physical systems they are not so closely connected. Accord- 
ing to the definition of the momentum vector usually adopted in quantum 
theory and reached later in this book, the components p^ are not necessarily 
algebraic quantities; they may be matrices or general symbols. On the 
other hand the components^ of the stream vector are necessarily algebraic 
quantities. 

The result (5-34) leads us to a new view of the wave equation. Consider 
a physical system described by a pure (i.e. factorisable) wave tensor /, or 
by the equivalent set of space tensors. We are not given the whole set of 
space tensors, but only one of the space vectors (j^ , j 2 , ^ 3 , jj together with 
the two invariants ^5 ( = 0) and j lB . We cannot therefore determine definitely 



66 Wave-tensor Calculus [5-3 

the factors of J; but our data are sufficient to limit them to certain possi- 
bilities, viz. they must be solutions of (5-34) and of the corresponding 
equation for #*. 

The wave equation is therefore an equation for determining the possible 
factors of a wave tensor, which is only partly known. 

At this point we must try to make clear a difference in our attitude towards 
wave mechanics from that which appears to be usual among quantum 
physicists. It will probably be agreed that wave mechanics is a method of 
analysis, not a theory of phenomena. The waves have no objective exist- 
ence; we invent them as required in solving our problems. In the present 
treatment we have found that any space vector can be expressed as a wave 
tensor. Ordinarily it is not a pure wave tensor; but it can be represented as 
a sum of pure wave tensors, which are then resolved into their wave-vector 
factors. If the space vector is a function of the coordinates, the wave-vector 
factors become "wave functions". In this way wave functions appear in 
connection with any characteristic of a system which is described by space 
vectors. It is therefore ambiguous to speak of the wave functions of a 
system; we should rather speak of the wave functions associated with some 
specified tensor of the system. Reference to the wave function or the wave 
equation of a system leaves us in the same state of conjecture as if reference 
were made to ' ' the tensor of the hydrogen atom " or " the equation of the sun ' ' . 

I am not cavilling at expressions, whose meaning is doubtless made plain 
either by the context or by custom. My point is that when wave analysis is 
our standard procedure when the ordinary tensor calculus is replaced by 
wave-tensor calculus we shall introduce new wave functions as casually 
as we introduce new tensors. The domain of physics treated in this book is 
for the most part different from that which has occupied the attention of 
writers on pure quantum theory. Sometimes our wave functions will 
coincide with theirs; sometimes they will differ. We find it well to maintain 
a certain amount of contact in order to utilise well-known results; but in 
principle we do not bind ourselves to use the wave functions that the 
quantum physicists have discussed. Remembering that the introduction of 
wave functions is merely a factorisation, we must obviously retain freedom 
to employ factorisation whenever it is useful. 

The reader must therefore be prepared to find here a greater elasticity in 
the definition and use of wave functions than he has been accustomed to. 

5*4. The Wave Equation as an Identity. 

If we represent E^E^E^E^ E 5 by the special pentad of matrices (3-27), 
it is not difficult to prove by straightforward verification that 



5-4] The Simple Wave Equation 67 

where is any four-valued quantity. Here, as usual, E^ is the four-valued 
quantity formed by chain multiplication, and (E^)^ is one of its four 
components, i.e. (E^) QL ^(E^) QL ^. 

Any other pentad E^ is obtained by a transformation E^ = qE^q- 1 . Let 
$' =g^; then E'^qE, so that 



Hence, multiplying (5-41) by q aoi q T p, we have 



it** 5 

S <JV*%(Wr- (# 16 f U# 16 0')r==0. (6-42) 



We can choose ty arbitrarily; because (since q is not singular) the corre- 
sponding ^ = y 1 ^' can be employed in (5-41). 

Thus the identity (5'41), verified for a particular pentad, is true for any 
pentad of matrices whatsoever. 

Let x* be another arbitrary four- valued quantity. Multiply (5-41) by 
initial x (inner multiplication). We have by (3'37) 



where 0* J %j E . The result is therefore 



or (ttiJi + E 2 h + # 3 J 3 + Et Jt + EsJ 5 - JP 16 j le ) = 0. (5-43) 

We can show similarly that 

. (5-44) 



Except that the term in j 5 is included, these are the wave equations as given 
in (5-34). They are here obtained as an identity satisfied by any two wave 
vectors 0, x* and their outer product J. 

We see that Dirac was right in restricting his hamiltonian to the above 
terms, instead of employing the sixteen terms of a complete space vector. 
By omitting j & he restricts the equation to stream vectors which have zero 
component normal to space-time; otherwise his equation is a perfectly 
general one satisfied by the factors of any pure wave tensor. The hamiltonian 
H is part of the complete stream vector J, except that the sign of j 16 is 
reversed. It must not be supposed that the components of J (other than 
j 5 ) which do not appear in H are zero. 

On the other hand Dirac's postulate that i/ and x* (or rather a quantity 
<* easily derived from x*) are conjugate complex quantities would restrict 



68 Wave-tensor Calculus [5-4 

J to very special forms. The restriction has to do with certain special 
applications, and is inappropriate in general theory. 
Multiply (5-43) by initial x* and apply (3*37). We obtain 

Ji 2 +J2 2 + Js 2 + J 4 2 + J 5 2 ~ Jie 2 - 0. (5-45) 

Again, multiply (5-43) by initial x*^& an( l a Ppty (3-37). We obtain 

JJlS +J2J2S +J3JM +JJ*5 = 0. (5'46) 

These, and the corresponding equations obtained by substituting other 
pentads, are relations satisfied identically by the components of a pure 
wave tensor. There are, of course, no such relations between the components 
of a general wave tensor T which is not stated to be factorisable. 



Let P a =j- 1? (a = 0, 1, 2, 3, 4, 5; M ^a). (5-47) 

/i=0 

For fixed a, the matrices E^ form a pentad. We therefore call P a a pentadic 
part of J. There are six pentadic parts which overlap, so that 

P a P a + qsJ = J. (5-48) 

The pentad which we have been using corresponds to <x = 0, and the wave 
equation (5-43) can be written 



But since the proof holds for any pentad, we have more generally 

CP.-tfieW^O (5-491) 

or equivalently P^iff = (qs J) iff (5-492) 

for all six values of a. 

From the present standpoint the use of the wave equation is to determine 
the factors of a pure wave tensor J. It seems to be generally true that in 
physics we determine a factor iff, not because its value is of particular im- 
portance to us, but because that happens to be the most convenient way of 
ascertaining that a factor exists, i.e. that J is pure. For example, the wave 
function iff of a hydrogen atom is investigated primarily because the mere 
existence of such a function imposes certain conditions on the hamiltonian 
(which is part of J), and these conditions determine the energy levels of the 
atom. It would probably be difficult to solve the more complex problems 
of quantum theory without evaluating iff, but since the observables of 
physics are always space tensors and therefore derived from wave tensors 
of the second (or higher) rank, the wave vector factors must ultimately be 
recombined. 

In the present case (which is perhaps too elementary to be typical) the 
conditions for purity of J are expressed directly by the equations (5-45) 

*<M\\ n.nd t.Viprp is Tin npari t,n pvalnat.** t.ViA fnnf.nra 



5-5] The Simple Wave Equation 69 

5*5. Standard Forms of Pure Wave Tensors. 

Equation (5*492) asserts: 

A factor of J is an eigensymbol of every pentadic part of J, and the eigenvahte 
of a pentadic part is qs J. (5-51 ) 

Consider an antitriad E^, E ar , E^ p (2-36). Each pentad contains one and 
only one member of an antitriad. Hence, in the expression (E^ v + E^ + EX P ) m, 
each term is a pentadic part and has eigenvalues im. Accordingly the form 

J^(E tLV E <jr E^ + E^)m (5-52) 

will, if the signs are properly chosen, satisfy the condition (5-51) that the 
eigenvalue .of every pentadic part is equal to the quarterspur. It turns 
out that four of the combinations of sign make J factorisable, and four 
do not. For example, take. the + sign for the first two terms; then 



Hence, if /*, v, a, T, A, p is an even permutation of 0, 1, 2, 3, 4, 5, 



so that the + sign must also be taken for the third term in order to satisfy 
(5-492). It is then easy to verify that J is factorisable by working out the 
factors in a particular matrix representation, or by testing it for idempotency 
according to the theory given in the next section. 
Accordingly our result is that 

J = (E^ + E m + EX P + j0 16 ) m (5-54) 

is a pure matrix if //,, v, or, T, A, p is an even permutation. Any two of the first 
three terms can be given negative sign, since this is equivalent to reversing 
the order of their suffixes and the permutation remains even. 

Any non-degenerate pure wave tensor can be reduced to the standard 
form (5-54) by a relativity transformation J' = qJq'. We first make a trans- 
formation so that one of the components, say J 5 , becomes zero. Then, by 

(5'45), 



Since J is non-degenerate, J 16 ^0. Hence the vectors (j l9 j 2 , fa, jj and 
Ois * ,?25 J35> ^45) have the same non-zero length, and by (5-46) they are at 
right angles. We can therefore choose two of the axes in four dimensions to 
coincide with them; we then have 



l = J25 = Jie > 32 = h = 4 = 15 = 35 = 

Applying (5-492) with <x = 0, 5, 3, we now have 

#1J10 = ih**l>> E U>3tot = ijltfr (^31 hi + ^32 J 

(5-552) 

Thus is an eigensymbol of the two commuting symbols E 19 E 25 and there- 
fore of their product iE u . It is therefore an eigensymbol of ( jE7 31 j 31 -f E Z2 j 32 ) ; 



70 Wave-tensor Calculus [5-5 

and the eigenvalue must be zero because (E 3l j 3l + E 32 j 32 ) anticommutes 
with 27 34 . Hence the third equation of (5-552) breaks up into 



so that ./si =#32* JH* 

Hence j 31 (E 3l iE^) $ 0; or, multiplying by E 9l , J 31 ( - 1 iE^) = 0. Now 
cannot be an eigensymbol of Jff la , because it is an eigensymbol of E^ which 
anticommutes with E 12 . Hence 

Jn = 0, J 32 = 0. 

Similarly we find J 41 = 0, j" 42 = 0. 

The pentad <x= 1 then gives (E^ + E^j^if/^ij^. Hence, by the first 
equation of (5-552), ^ 12 = 0. All the components are now accounted for, and 
J reduces to (E l -f E% 5 + E M + E I6 )j^ , which is of the required standard form. 

To obtain a standard form for a degenerate pure wave tensor we proceed 
as follows. Let J = 0x* be a degenerate pure wave tensor (J 16 = 0), and let 
jvr be a component which does not vanish. Then E m J is a pure wave tensor, 
since it has factors E w and #* ; and it is non-degenerate since its quarterspur 
is E w (E^j^) = j v . Hence E W J can be reduced to the form (5-54) by a 
relativity transformation. We take therefore 



E v cannot be E^, E OT9 E^ or B 16 , since J would then be non-degenerate; 
but it can be any other U-symbol. Taking E w = E^ , we obtain 

J=(E va + E^iE VT + iE^)m. (5-56) 

This is the standard form for a degenerate pure wave tensor. 

5*6. Idempotency. 

A symbol J is said to be idempotent if J 2 = J. 

To normalise an JS?-number we multiply it by an algebraic factor so as to 
make the quarterspur J. If it is represented as a matrix (so that a spur 
exists) we normalise it by making the spur 1. It is, of course, impossible to 
normalise a degenerate ^/-number. 

We shall show that a necessary and sufficient condition for a non- 
degenerate matrix to be pure is that it shall be idempotent when normalised. 

(5-61) 

Let J = ^x* be a normalised matrix so that 



spur J= 

Then J 2 =^X*^* = ^- 1 -X* SS= ^ 

so that the condition is necessary. To prove that it is sufficient, let T be a 
matrix satisfying ^^ spur y =L ., 



5-6] The Simple Wave Equation 71 

Any matrix can be expressed as the sum of a number of vector products; 
therefore let .... (6-63) 



Here the suffixes a, b, c, ... distinguish different vectors, the row-and-column 
suffixes being omitted as usual. We write A^ for the scalar product Xa*^* 
so that the product *l* a Xa*'l l bXb* reduces to -^a&^aXb*- Then 

T i -2 f -(^-l)^Xi.* + ^*a* + ^^X-i* + --0. (5-64) 
Corresponding to the four suffixes of x this gives four linear equations satis- 
fied by the vectors $ a , & , */t c , .... Using any one of these equations to give 
the value of a in terms of the other ^'s, we can eliminate ^ ft in (5-63) and 
so reduce by one the number of vector products on the right-hand side of 
(5-63). Repeating the process, we reduce the number of vector products one 
by one. 
The procedure fails if the coefficients of ifj a vanish in all four equations, 



But we can then use (5-65) to eliminate x* in (5-63), and the number of 
vector products is again reduced by one. 

The reduction can be continued so long as there are any non-vanishing 
coefficients in (5*64). When all the coefficients vanish so that 

A a a ~ AM ^ ACC ~ = *> A a fr = **ba ^ = ", 

no further reduction is possible. We then have, by (5-63), 

spur T = spur $ a x a * + spur fa x & * + spur $ c x* + 



But spiir T= 1, so that there can be only one term on the right-hand side. 
That is to say, T is the product of two vectors. 

A pure matrix is necessarily singular. This follows from 3-7 (6), since the 
idempotent condition, 7 2 J = 0, gives eigenvalues and 1. A singular 
matrix is not necessarily pure. 

If the square of a matrix is 1, the question sometimes arises whether it 
has to fulfil any other condition in order that it may be a member of a com- 
plete orthogonal set. The most commonly occurring combinations whose 
squares are algebraic are triadic and pentadic expressions; these can serve 
as individual members of a new complete set. But it has been pointed out 
by D. E. Littlewoodf that we can also form combinations of antiperpen- 
dicular matrices whose squares are algebraic, and these cannot be members 
of a complete set. We find by direct multiplication that the square of 

(5-66) 



f Journ. Land. Math. Soc. 0, 41 (1934). 



72 Wave-tensor Calculus [5-6 

is 1. Of the eight possible combinations of sign in an antitetrad, four yield 
factorisable matrices, as we have seen; the other four give matrices whose 
squares are algebraic. 

The quarterspur of (5-66) is \i, whereas the quarterspur of E^ is or i. 
Since the quarterspur is invariant for the transformation F fJlf = qE tl q f 9 there 
can be no such transformation connecting (5-66) and E^. Therefore (5-66) 
cannot be a member of a complete set. We shall call an expression of the 
form (5-66), or reducible to it by a relativity transformation, a compact 
U-number. 

I have not as yet found any physical application for compact JS7-numbers; 
but perhaps others will be more successful. They surely must have an im- 
portance of some kind, possibly in the theory of radiation or even in the 
theory of the nucleus subjects which we do not seriously attempt to treat 
in this book. 

5-7. Spectral Sets. 

We suggested in 5-4 that the wave vector was investigated in physics, not 
for its own sake, but because the existence of factors imposes certain in- 
variant conditions on the stream vector and on the hamiltonian which forms 
part of it. We may now go a step further, and say that the condition which it 
is sought to impose is that of idempotency. Those familiar with the Group 
Theory of wave mechanics will recall the fundamental part played by idem- 
potent operators in selecting the "pure" states of a statistical ensemble. 
Consider the wave tensors 

J fl =-ii (tf 



We have found in (5-54) that these are pure. Since the quarterspur is 
Ji7 16 = J, they are normalised. Hence they are idempotent, as can be 
verified by direct multiplication. We can also verify that their products are 
zero. They accordingly satisfy 

A set of operators satisfying the conditions (5-72) is called a spectral set. 
Here the set consists of four operators only. The more familiar examples of 
spectral sets in physics includean infinite number of operators. For example, 
let J\ denote the operation of selecting light of wave length A from a source 
of light represented by $; thus the light of wave length A existing in the 
source is represented by J A ^. If we repeat the selective operation J\ on 
J A ^, it makes no difference; hence J^ = J\. The symbol ^e^ denotes the 
operation of selecting wave length A' out of light already selected as being 



5-8] The Simple Wave Equation 73 

of wave length A; the result is obviously zero. Further, selecting every wave 
length in turn and adding the results, we reproduce the original source of 
light; hence S A JA is equal to the "identical operator" 1. The selective 
operators of spectral analysis therefore fulfil the equations (5-72), which 
ensure that they are idempotent, non-overlapping and exhaustive. 

As G. Temple has pointed outf it is equations of the form (5-72) which 
directly embody the physical conception of a "pure constituent". The 
mathematically convenient criterion of purity, namely factorisability of 
the operator in matrix representation, should be regarded as derived from 
(5-72) rather than vice versa. 

This suggests a new approach to the theory of the representation of 
phenomena by -E-symbols. We can regard the matrices E^ as introduced 
by a spectral analysis of entities represented by algebraic numbers (in 
particular, probability distributions or densities) into four pure constituents 
given by (5-71). This point of view is developed in 13-0. 

5*8. The Complete Stream Vector of a Particle. 

Consider a particle in spherical space-time. A classical particle is described 
by two 4-vectors, namely a position vector and a velocity vector. In five- 
dimensional representation the position vector is the radius of space-time 
which passes through the particle; the velocity vector is at right angles to it 
and lies in the four-dimensional hypersphere. 

If we take axes such that the position vector is in the E & direction and the 
velocity vector is in the E direction, the two vectors reduce to single com- 
ponents j 5 and .7*4 . Let us treat them as components of a single wave tensor. 
There is, of course, no compulsion to combine them; there is no unique 
definition of the wave tensor of a particle, any more than in ordinary rela- 
tivity theory there is a unique definition of the tensor of a particle; and it 
would be legitimate to investigate a tensor representing position only or 
velocity only, if desired. But we shall try to find a tensor, called the complete 
stream vector, which comprises both. 

If the complete stream vector is pure, it must have two more com- 
ponents besides j" 4 andj'g. We may take it to be 

(^41 + ^50 + ^40 + ')> ( 5 * 81 ) 

which is of the standard form (5-56) with p, v, a,r = 5, 4, 1, 0. The additional 
terms define an axis in the three-dimensional space, which is in some way 
characteristic of the particle. This axis, which in (5-81) is taken to be in the 
E l direction, is called the spin axis. 

The question now arises whether in ascribing a complete stream vector J 
to the particle we should take (5-81) to be the actual vector J or the vector 

t "The Physical Principles of the Quantum Theory", Proc. Roy. Soc. A, 138, 479. 



74 Wave-tensor Calculus [5-8 

density J . iE 6 . This is answered by the Uncertainty Principle, which asserts 
that a particle cannot have exact position and exact velocity simultan- 
eously. Thus our combination of a position vector and a velocity vector will 
not apply to a discrete particle, but describes an element of its probability 
distribution. We must therefore take (5-81) to be a vector density, so that 

t JE 6 = (E^ + EM + iE M + iE^) a. (5-82) 

From this we obtain J = (E n + E 19 + JB7 46 + E ol ) a, (5- 83) 

which is of the standard form (5-54) for a non-degenerate pure tensor. 

The direction of the spin axis is shown by the term E ol , or equivalently by 
the term E 23 which gives the plane of the spin. The velocity vector, which by 
our choice of axes is in the time direction, is represented by the term E^, 
this being the matrix of the rotation which would displace the particle in 
the time direction. Instead of a position vector, we have an invariant E 19 . 
The "position" of the particle is therefore invariant for all relativity rota- 
tions; this is only possible if we represent the particle as an entity uniformly 
distributed throughout the hypersphere of space-time. This is in agreement 
with the uncertainty principle; for we have ascribed an exact velocity 
vector J57 46 to the particle, and therefore its position is entirely indeterminate. 

The attempt to assign a combination of position vector and velocity 
vector to a particle breaks down, as the uncertainty principle foretells. The 
position vector H7 60 in (5-82) defines, not the position of the particle, but the 
position of an element of its probability distribution selected for con- 
sideration. 

We define an elementary particle to be an entity whose characteristics are 
completely specified by a complete stream vector of the type (5-83), so that 
it can be represented by simple wave vectors 0, x*~ It has exact momentum 
but indeterminate position. This would perhaps more usually be called an 
elementary state of an elementary particle; and it is contemplated that a 
number of elementary states may be superposed forming a wave packet 
which has approximate position and momentum. It is to be remembered, 
however, that the properties of observational significance are relations to 
other elementary particles or combinations of particles, and not the primitive 
relations to a symbolic frame summed up in (5'83). We must not be in too 
great a hurry to identify our formulae with those employed in the practical 
applications of quantum theory. 

By (4-65) the three-dimensional vector density or strain vector corre- 
sponding to J is S^iJEto* For the special wave tensor (5-83), we find 

S=-J. (5-84) 



CHAPTER VI 
REALITY CONDITIONS 

6*1. Distinction between Space and Time. 

In relativity theory the interval between two point-events is defined by its 
square ds 2 . In Galilean coordinates 

ds* = dt* - dxf 



the velocity of light being taken to be unity as usual. By the use of anti- 
commuting symbols, the square root can be expressed in rational form; thus 
we may write 



On squaring, the product terms cancel owing to the anticommutation, and 
we have (rfg)2== _ dx *_ dx *_ dx *_ dx * = ds * 

if a; 4 = it. The algebraic square root ds is the eigenvalue of ds. Since (6- 1 1 ) 
is the space vector, or displacement, between the two points, our conclusion 
is that the interval is the eigenvalue of the displacement. 

It is of fundamental importance that, since a pentad contains three 
imaginary and two real matrices, we cannot cover more than three dimen- 
sions with matrices of the same real or imaginary character. If then we use 
the imaginary matrices E l9 E 2 , E% for the three similar space dimensions, we 
have to use a real matrix E for the fourth dimension of the physical con- 
tinuum of equivalent points. Thus the distinctive character of the fourth 
dimension (time) is already foresJiadowed in the constitution of the pentads. 

For real phenomena x 1 , x 2 , x 3 are real, and a: 4 ( = it) is imaginary. Since 
E 19 E 2 , E 3 are imaginary and E^ is real, the vector interval ds is a wholly 
imaginary matrix. 

A matrix which is wholly real or wholly imaginary will be called mono- 
thetic. Two matrices are homothetic if both are real or both imaginary, and 
antithetic if one is real and the other imaginary. The distinction between 
space and time is comprised in the statement that ds is a monothetic matrix. 

In Schrodinger's wave mechanics certain "Hermitic conditions" were 
imposed in order that the mathematical expressions should correspond to 
real physical phenomena. It seems to have been generally assumed that 
the reality conditions of Dirac's wave mechanics must be of the same 
Hermitic form, although several writers have pointed out the difficulties 
arising from this assumption. In my own developments, I abandoned 
Hermitic conditions at the outset; it seemed illogical to retain them in an 
analysis which recognises sixteen different square roots of 1. We shall 



76 Wave-tensor Calculus [6-1 

determine the reality conditions of the present theory independently. By 
relativity considerations it is possible to determine them uniquely. 

For a formal treatment, it is best to begin by considering the reality 
conditions for rotations. Corresponding to a matrix E^, we can consider 
two antithetic rotations, E^Q^, E^iu^, where 0^ and u^ are real. Only one 
of these will be admissible for physically real phenomena. Now "physical 
reality" is an invariant property; therefore our reality condition must be 
such that it is invariant for all relativity rotations of the frame of reference, 
provided that these rotations are themselves physically real and therefore 
satisfy the reality condition that is being considered. 

For trial suppose that a rotation q = e^ 6 ^ is physically real, if E^O^ is 
imaginary. Then q is in general complex. Consider another rotation 
q l = e \E v Q v ^ w h ose matrix E V 9 V is imaginary and therefore satisfies the pro- 
posed reality condition. If now we apply the physically real rotation q to the 
frame of reference, E v 6 v -^q(E v 6 v ) q'. IfE^ anticommutes with E v , q (E V 8 V ) q' 
is complex, so that q t no longer satisfies the reality condition. Thus the 
proposed reality condition is non-invariant, and must be rejected. 

Accordingly the reality condition for a rotation is that q must be real. 
Then the matrix E V Q V of the rotation q l is real, and remains real when it is 
transformed to qE v 6 v q' by the real rotation q. 

The essential point in the argument is that the matrix of a relativity 
rotation can be transformed by applying another relativity rotation; so that 
any proposed reality condition is employed twice over and its self-consist- 
ency is thereby put to a test. In other words we have to secure that the 
physically real rotations constitute a Group. 

Considering the most general rotation in four-dimensional space-timef 
g = exp I (E 23 2 s + E 3l 31 + E l2 e i2 + Euiu u + Euiuu + E^ (6-12) 
the condition that q is real requires that 23 , 31 , 12 , ^ 14 , % 24 , ^ 34 shall all be 
real. An E u rotation gives, as in (--15), 

Xi = x l cos (iu u ) - 4 sin (m 14 ), x = x l sin (iu u ) + # 4 cos (iu u ), 
or, if # 4 = it, 

Xi = x l cosh w u + 1 sinh u u 9 t f = x l sinh % 4 + 1 cosh w 14 , (6-13) 
so that the rotation is hyperbolic (Lorentz transformation). Thus the rela- 
tivity transformations in four dimensions consist of three circular rotations 
and three Lorentz transformations u in agreement with experience. 

As a by-product we see that the real quantity concerned in the Lorentz 
transformations is t = xji, so that a 4 is imaginary (ifx l9 x 2 , x 3 are real). We 
have thus a deductive proof of the result, already noticed, that (6-11) is a 

t We here exhibit the six elementary rotations collected together for reference. As 
explained hi 4*2, it would be necessary to restrict them to infinitesimal rotations if it were 
intended to apply them simultaneously. 



6-2] Reality Conditions 77 

monothetic matrix. It is a matter of convention that it is imaginary, not 
real. By taking x l9 x 2 , x 3 real, so that # 4 is imaginary, we conform to the 
convention of ordinary relativity theory which assigns real measure to 
time-like intervals. But we might equally have taken x l9 x 2 , x$ imaginary 
so that # 4 would be real; ds is then real for space-like intervals. Thus the 
compulsory reality conditions are: 

For rotations (in four dimensions): q is a real matrix, 
For intervals or displacements: ds is a monothetic matrix, 
Alternatively we may measure an interval by its vector density 



) 

V 

ix, j 



Then by (6-11) -ids^E^dx^E^dx^ + E^dx^ + E^dx^. (6-15) 

Since E 5 is real, ds is antithetic to ds; and with the usual convention ds is 
real and ds is imaginary. 

At this stage it is well to review the progress of our theory of space and 
time. We have shown that, starting with a basal wave vector ^r, it is possible 
to construct a continuum of "equivalent" points which forms a four- 
dimensional hypersphere in five dimensions. In Chapter iv we showed that 
this continuum has the local isotropic quality of ordinary space-time in 
that rotations in any of its coordinate planes are relativity rotations; but 
we did not there discriminate between circular and hyperbolic rotations. We 
have now confirmed the resemblance in greater detail by showing that the 
relativity transformations of this theoretical continuum consist of three 
ordinary rotations and three Lorentz transformations; or equivalently that 
one of the four dimensions is antithetic to the other three. 

Further, referred to this continuum, the basal vector $ has the trans- 
formation properties of Dirac's $, and in fact satisfies identically an equation 
identifiable with Dirac's wave equation. 

We must remind ourselves, however, that this is no more than the embryo 
of the actual macroscopic space-time of our experience. In the next section 
we shall find a very significant difference which shows the need for intro- 
ducing further developments in due course. 

6* 2 . Translations . 

Considering the neighbourhood of a particular point P on the hypersphere, 
we take as usual the coordinate x 5 to be along the radius at P, so that 
#1, #2 #3 #4 are rectangular coordinates in space-time. This coordinate 
system is necessarily local, for it is impossible to construct an extended 
system of rectangular coordinates in a curved space. We must therefore 
restrict ourselves to an infinitesimal region around P. 

The transformation q = e^ E ^ gives a rotation of the hypersphere in the 
plane x l x^ and therefore displaces P in the x l direction. Thus infinitesimal 



78 Wave-tensor Calculus [6-2 

translations of a point along the four axes in space-time correspond to 
relativity rotations with matrices U 16 , E 2b , JS? 35 , U 45 . 

The recognised relativity transformations in space-time, viz. four trans- 
lations, three rotations in space and three Lorentz transformations, make 
up the ten relativity rotations in five dimensions introduced in 4'3. The 
customary approximation which treats space-time as flat hides the fact that 
translation is a form of rotation, viz. rotation about the centre of curvature 
of space-time. 

The general translation in space-time is accordingly given by the trans- 
formation 9==exp KEuOu + E n O n + E n O u + EM. (6-21) 

If we apply the previous reality condition that q is real, the expression in 
real variables is 

q = exp (E 15 iu lb + J 25 m 25 + E^iu^ + E^), ( 6 ' 22 ) 

since E 15 , jE 25 , J57 35 are imaginary, and E^ is real. As in (6-13) the u's give 
hyperbolic rotations, and 45 gives a circular rotation. This means that the 
continuum is open in three (space) dimensions and closed in one (time) 
dimension the reverse of the conditions in actual space-time. 

The root of the trouble is that, when space-time is pictured in five dimen- 
sions, these dimensions are four space-like and one time-like; for the radius 
of curvature represented in the fifth dimension is space-like. But the matrices 
associated with these dimensions are three imaginary and two real. The real 
matrix J57 5 is incongruously associated with a space-like dimension. 

Some writers on relativity have mooted the possibility that the world 
might have negative curvature. (Negative curvature refers to the Gaussian 
curvature which is proportional to l/B 2 . If B 2 is negative, B is an imaginary 
length, or equivalently it is a t'me-like radius.) But the proposal has been 
treated from the point of view of formal mathematics, and can scarcely be 
entertained in physical theory .f 

We shall find the significance of this incongruity in the next section. But 
we shall first consider how it is to be reconciled with the invariance of reality 
conditions. To conform to the actual universe it will be necessary to admit 
that the exponent of q is real in (6-12) but imaginary in (6-21); that is to 
say, the general rotation is to be resolved into = Q l + 2 , where 

i = + ^8i + ^iiii + ^i4^4 + ^i4^+^^il 

2 = JS? 15 15 + -^25^25 + -^35 ^35 + ^45 *^45 > I 

with the reality condition that X is real and 2 is imaginary. 

What then becomes of the invariance of physical reality? The saving 
circumstance is that we have restricted ourselves to an infinitesimal region 

t The objection to unclosed space arises from quantum theory rather than from relativity 
theory. It will become obvious as our investigation proceeds. 



6-2] Reality Conditions 79 

round a point P\ and at the point P the direction denoted by E 6 (normal to 
space-time) is absolutely distinguished from the other directions (4-4). 
Thus the separation of into X and 2 is defined in an absolute way; and 
we can attach different reality conditions to the two parts without coming 
into conflict with relativity principles. 

Let us now apply a rotation 1 ' + 2 ' to a rotation 1 + 2 . Under the 
transformation g 1 = ei e i', x and 2 transform separately. By the above 
reality conditions q l is real, and therefore ft!?/, ?i0 2 ?i' remain real and 
imaginary respectively, and continue to satisfy the reality conditions. 
Under the transformation g 2 = e ie ', Q l and 2 are not kept separate. The 
real matrix l is transformed into a complex matrix g 2 1 ? 2 / ; but it is easily 
verified that the imaginary terms in it are of the form 2 so that they 
satisfy the reality conditions. Similarly the imaginary matrix 2 becomes 
a complex matrix ? 2 2 g 2 / ; but the real terms introduced are of the form 
X , and satisfy the reality condition for lB Thus the reality conditions 
are found to be self-consistent, and the rotations which satisfy them form 
a Group. 

We have confined ourselves to an infinitesimal (initial) region because it 
is only locally that we can pick out a unique direction, absolutely different 
from other directions, to distinguish as J? 5 . But is it sufficient to treat an 
infinitesimal region? A relativity rotation cannot be real at some points 
and unreal at other points of space-time; we ought therefore to show that 
the same rotation tested in two different localities fulfils the proposed 
reality conditions at both or neither. It turns out that this self-consistency 
is assured automatically. If we examine the reality condition at another 
point of space-time, we must first make sure that it is a real point; and the 
test of its reality is that it is equivalent to the real point first considered 
that it can be transformed into it by a relativity rotation which satisfies the 
reality condition. Thus we have to lay down the reality condition for rela- 
tivity transformations of a single initial point (given as real) before we can 
decide what values of the coordinates represent real points, i.e. points which 
have real equivalence to a point known to be real. Thus we have not to show 
that our adopted reality condition is self -consistent over a predetermined 
real domain; the domain over which it is self-consistent is ipso facto the 
domain of real points. 

We begin with one real point P the observer, in fact, for all reality is 
relative to him. We determine a group of infinitesimal transformations which 
we define as the physically real transformations. These transformations 
applied to P give all the neighbouring real points. By a process of continua- 
tion (using at each stage the local reality conditions) we reach the more 
distant real points. Any of these real points can now be taken as the initial 
real point; proceeding from it we shall by the same construction obtain the 



80 Wave-tensor Calculus [6-2 

same real domain. This is a consequence of the group property of the 
transformations used. 

We have shown that the reality conditions (6-23) satisfy relativity re- 
quirements; but we have still to explain why these are the conditions pre- 
vailing in nature, rather than the simple condition that Q l and 2 should 
both be real. 

6*3. Neutral Space -time. 

We shall find later that the entity represented by a simple wave tensor is an 
electric particle (proton or electron). Up to the present we have dealt with 
one particle only; we have not yet developed the apparatus of description 
for two or more particles. It is therefore rather premature to talk about 
space-time, which is a macroscopic conception presupposing vast numbers 
of particles. What we have investigated is a preliminary geometrical frame- 
work in which the characteristics of a single elementary particle are repre- 
sented vectorially . This framework is the genesis of macroscopic space-time; 
and it already contains two of the most essential features: (1) four dimen- 
sions, of which one is antithetic to the other three, and (2) a radius of 
curvature. 

For clearness we shall here anticipate some of the changes which will take 
place in converting this preliminary conception into the space-time of 
macroscopic experience. By introducing great numbers of particles the 
radius of curvature will be greatly increased, relatively to the linear scale 
characteristic of a single particle (commonly recognised as the wave length 
of its Schrodinger waves). The increase of the population will give scope for 
irregularity, and the hypersphere of space-time will be distorted by gravi- 
tational fields. But the difference that chiefly concerns us now is that there 
will be a balancing of positive and negative particles. A universe containing 
only one particle, and therefore only one sign of charge, is lop-sided com- 
pared with a universe containing equal numbers of positive and negative 
charges. It will readily be imagined that if matter consisted of electrons only, 
the enormous negative potential would so alter the world that we should 
require a different type of space-time to frame the phenomena. It is just 
such a world in miniature that our theory of a single charged particle 
imitates. 

The Riemannian space-time of Einstein's general relativity theory is 
derived from the extensional relations of neutral matter. The test bodies 
whose behaviour determines its characteristics scales, clocks, moving 
particles, light waves are electrically neutral. Moreover, all macroscopic 
matter in our experience is to a very high approximation neutral; if the 
proportion of electrons to protons differs from equality by one in a billion, 
the electric charge expressed in ordinary units is stupendous. 



6-3] Reality Conditions 81 

The test bodies are used to measure intervals, and hence determine the 
tensor g^ v characteristic of neutral space-time. Ideally their use is restricted 
to regions where there is no electromagnetic field. In practice this is not a 
very important restriction, because the strongest electromagnetic fields 
encountered in nature correspond to a very trifling lack of balance of dis- 
tribution of positive and negative charges. But in an electromagnetic field 
too strong to be neglected, it would be impossible to use the test bodies to 
determine g^ v , because there is no agreed definition as to how g^ v is related 
to the indications of the test-bodies when electromagnetic fields are present. 
Each investigator has defined it according to his own fancy. No experi- 
menter would undertake to make accurate measurements of length in an 
intense electromagnetic field; statements about lengths and distances 
within the field are inferences from observations made outside the field, and 
depend on the theoretical formulae employed in calculating the inference. 
Current scientific literature abounds in rival formulae (usually embodied in 
an "action-principle") for making such calculations, each corresponding to 
a different definition of g^ v in regions where there is no means of determining 
it by direct observation.! 

We must therefore consider even at this early stage the main difference 
between the space-time of a universe whose content is neutral and the 
space-time of a universe containing a particle or particles of one sign only. 
We shall call the space-time of a universe containing only positive particles 
positively saturated; if containing negative particles only, it is negatively 
saturated. 

In 3-9 we have shown that there exist two kinds of frame, right- and 
left-handed, which cannot be transformed into one another by a relativity 
rotation. If we take two vectors T = S ^ B^ , T' = S ^ F^ , where E^ , F^ are 
respectively right- and left-handed, these cannot be transformed into one 
another by a relativity rotation. Clearly a distinction of this kind is 
required to discriminate between positive and negative charged particles. 
We shall therefore provisionally identify T and T' with the complete stream 
vectors of positive and negative particles subject, of course, to confirma- 
tion by detailed examination of the Resulting properties. 

Take right- and left-handed frames related as in (3'92), so that 

^18> ^5> ^15 ^25> ^35> ^45 = ~"^16> ~^5 "~^15 ~^25> ""^35 ""^45- 

(6-31) 
For the second particle we have T' = S t^ F^ = X t^ E^ , where 

^18 *5 W* ^25' W> ^45 ^ "~"'l6> ~~^5> "~^L5 ~~*25 ~^35 "~^45 (6'32) 

the other ten components being the same for both particles. Hence, con- 
sidering the position vector, the coordinates of the two particlesj are 

t See 13-4. 

j More strictly coordinates of elements of their probability distributions ( 5*8). 



82 Wave-tensor Cakulus [6-3 

respectively (^ , 1 2 , (, , 4 , J 6 ) and (^ , J 2 , * 3 , 4 , - J 6 ). If we take as usual the E 5 
axis normal to the small region that we are considering, so that t , t 2 , t 3 , t^ 
are infinitesimal, the particles are in antipodal regions in five dimensions; 
but the ordinary four-dimensional point of view is that they are at the same 
point of space-time (^ , t 2 , tf 3 , J 4 ), f and that the centre of curvature of space- 
time is in opposite directions along the normal, according to which particle 
we are considering. 

There is nothing surprising in this. The two particles are contemplated as 
alternatives; for our analytical machinery is not yet capable of dealing with 
two particles at once. By Einstein's theory the curvature of space-time 
depends on its contents. So, if we start by considering an infinitesimal region, 
we do not know how the region will continue until we have decided on the 
contents of the region. It will curve away from the tangent plane more or 
less strongly according as the density is high or low. We now see further that 
it will curve to one side of the plane or the other side of the plane according 
as we insert an elementary positive or negative charge (or a probability there- 
of). This is not noticed in Einstein's theory, because that is a theory of macro- 
scopic matter, which (even if it is electrically charged) contains equal num- 
bers of positive and negative charges to an extremely high approximation. 

It will be seen that the idea first suggested by the five-dimensional picture 
that the positive and negative particles are at antipodal points on a fixed 
sphere is rather misleading. They are not simultaneously present; and the 
sphere is not fixed until we have decided which is present. 

Our result is that, considering an infinitesimal region of space-time, if it 
contains part of the probability distribution of a positively charged particle, 
the radius of curvature will be in one direction of the normal, say # 6 ; and if 
it contains part of the probability distribution of a negatively charged 
particle, the radius of curvature will be in the opposite direction # 5 ; or 
briefly, if the region is positively saturated the radius of curvature will be 
in the direction # 6 , and if negatively saturated it will be in the direction 
# 5 . Suppose now that it is neutral having equal probability of positive 
or negative charge. Two possibilities are open. The curvature may be zero, 
or the radius of curvature may be in the direction of the imaginary normal 
ix b . These are the only alternatives which have neither a positive nor a 
negative bias. We know from ordinary relativity theory that the first 
alternative is incorrect; neutral matter does involve a curvature of space- 
time. Therefore we must accept the second alternative; the radius of curva- 
ture is in the direction ix 6 . That is to say the radius of curvature changes its 
character from time-like to space-like when we pass from a positively or 
negatively saturated world to a neutral world. 

t We have chosen the relation (3-92) rather than (3*93) or (3*94) in order that our formulae 
may refer to particles at the same point of space-time. 



Reality Conditions 83 

This result will be confirmed in our subsequent investigations. But I 
think it is clear, even at this early stage, that it is forced upon us. 

We can now understand the origin of the curious reality conditions in 6-2. 
The whole trouble was that the radius of curvature was associated with a 
real matrix E$ indicating time-like character. But the radius of curvature is 
time-like in the positively or negatively saturated world to which the most 
elementary formulae relate. It is in the transition to a neutral world that 
the space-like radius of curvature becomes substituted for a time-like 
radius. In a positively saturated world the relativity transformations 
q = e* satisfy the simple condition that is a real matrix. The V^l never 
gets a footing in so simple a world. According to this theory a positively 
saturated world is open in its space dimensions and closed in its time 
dimension. Saturation is so remote from the conditions of our actual ex- 
perience that we certainly cannot bring forward any observational evidence 
to the contrary. We can well imagine that the stupendous electrical repul- 
sions would be sufficient to burst any closed space. 

Since the vector E 5 x & towards the centre of curvature has in neutral 
space-time a value antithetic to its value in electrically saturated space- 
time, it is antithetic to the four-dimensional position vector 

X = E^ + E 2 x% + EzXz + E^x^ . 

The rotation about a centre at J 5 # 6 , which produces a displacement dX, is 
given by dX/E 5 x^ . Thus the rotation 2 is antithetic to its value in electric- 
ally saturated space-time, and is therefore imaginary. This gives the reality 
conditions (6-23). The change from a time-like to a space-like radius of 
curvature from saturated to neutral space-time is the source of the V^l 
which is such an inescapable feature of quantum formulae. 

6*4. Kinematical and Electrical Matrices. 

For general developments it is more convenient to take right- and left- 
handed seta Ep, Fp connected by (3'93),f so that 

^>^i6= -JV-^i6 (ft =1,2, 3, 4, 5), (6-411) 

the other ten matrices being the same. Let 

N^N^^iE^iE^. (6-412) 

We shall call N^ a neutral set. As a further generalisation we define a macro- 
scopic set Hp by ^ *-*%, A^, (6-413) 

the other ten matrices being the same. Then A is a scale constant which may 
be real, imaginary or complex. Real values correspond to electrically 
saturated space-time, and imaginary values to neutral space-time. An 

t We shall see in 6-5, that (3-92) gives the association of vector densities and (3-03) the 
association of vectors. 



84 Wave-tensor Calculus [6-4 

ordinary field of positive or negative potential could therefore be repre- 
sented by a complex value of A, the real part being very small compared 
with the imaginary part; this kind of representation is only used when we 
are pursuing a unified field theory, involving non-Riemannian geometry. 
More usually macroscopic electrical fields are represented as perturbing 
influences superposed on Einstein's neutral space-time. 

We may express a complete space vector in terms of macroscopic com- 
ponents, namely T^^m^M^^m^E^XL.m^E^ (6-42) 

where S fc refers to the ten unchanged matrices and S c to the six changed 
matrices in (6-413). We shall call the unchanged matrices the kinematical 
matrices, and the matrices whose sign is changed between right- and left- 
handed frames the electrical matrices. The electrical matrices are accordingly 

E l9 E 29 EZ, EI, E &9 J57 16 (6-43) 

and the kinematical matrices are those associated with the ten rotations in 
five dimensions. We shall adhere to this nomenclature, irrespective of the 
physical interpretation.! 

The relativity rotations in five dimensions correspond to the ten kine- 
matical matrices. We may therefore distinguish them as kinematical 
rotations, the other six relativity transformations being electrical rotations. 
As shown in 4-3 the kinematical and electrical parts of (6-42) are trans- 
formed separately by the kinematical rotations; so that from the ordinary 
standpoint they are separate vectors arbitrarily combined into one analy- 
tical expression. In combining them we can introduce a scale constant A, 
which remains invariant in the transformations. For example, in macro- 
scopic spherical space-time we may meet with two distinct but somewhat 
analogous vectors, say a velocity and a spin. We think it probable that there 
is some significant combination of these into a complete space vector, which 
will lead to a more far-reaching theory of the phenomena. But until the 
details of that theory are worked out, the combination involves an undeter- 
mined scale ratio A. If we regard the components m^ of the complete 
vector as definite and equal to the components of the two separate vectors 
as ordinarily measured, A must be embodied in the frame of our ordinary 
measurements which then becomes a macroscopic frame M^ . 

We have found representations of a complete space vector in five dimen- 
sions ( 4-3) and in four dimensions ( 4-4). There is also a representation in 
six dimensions. The components t l9 t 2 , ... t l& form a 15-vector, or anti- 
symmetrical tensor of the second rank in six dimensions. From a purely 
algebraic standpoint this is the most fundamental representation; and the 

f The matrix associated with a particular characteristic depends on whether the 
characteristic is expressed as a vector, vector density or strain vector. Thus the "electrical 
matrix" E l has not always an electrical significance. 



6-5] Reality Conditions 85 

theory of U-symbols is primarily a study of the group of rotations in space 
of six dimensions. But the ordinary vectors of that space (6-vectors) 
cannot be expressed in terms of -E-symbols. The physical application of this 
group is specialised by the fact that space, as defined in physics, is an 
abstraction of the extensional relations of neutral matter, and therefore 
involves the superposition of a right- and a left-handed frame in the 6-space. 
This superposition involves the selection of a pentad (here taken to be /5 0/t ) 
which, together with E IB , is reversed in sign between right- and left-handed 
frames. One suffix (in this case 0) thereby acquires distinctive properties; 
and, once chosen, it remains associated with an invariant direction in the 
6-space. The remaining five dimensions become the 5-space of 4-3. 

To review the argument: A right-handed frame can be represented as 
having complete symmetry in a (5-space. Such a frame would be appropriate 
as a reference system if the universe consisted of particles of one sign 
only. Since actual physical systems consist of nearly equal numbers of 
positive and negative particles, our actual reference system is based on a 
frame which is a superposition of right- and left-handed frames, the one 
being, as it were, a reflection of the other. In the composite frame it is 
no longer true that all directions in six dimensions are equivalent; one 
direction must be chosen as the axis of the reflection, and thereby becomes 
distinguished from the others. We have associated the suffix with this 
direction. Thus, whilst the elementary right-handed frame exhibits six- 
dimensional relativity, the actual composite frame exhibits five-dimensional 
relativity. Accordingly the starting point of the theory of actual space- 
time is the five-dimensional representation of wave tensors treated in 4-3. 

The kinematical rotations have the same relations to the frames E , F , 
Np , Mp . In each case they rotate separately the two parts into which (6-42) 
is divided, and the value of A does not affect the result. But the electrical 
rotations have opposite effects on E^ and F^ , and therefore correspond to 
some kind of separation of electric charge, or polarisation. For this reason 
they give apparently non-relativistic transformations of neutral or macro- 
scopic space vectors. This does not mean that it is unprofitable to investigate 
them further; but it is only by dropping the usual representation in Rie- 
mannian space-time, and following a unified geometrical representation 
of gravitational and electromagnetic fields, such as those of Weyl and the 
author, that these electrical transformations become admissible. 

6-5. Summary of the Reality Conditions. 

Since physical quantities can be expressed as space vectors, vector densities 
(four-dimensional), or strain vectors (three-dimensional densities), we have 
to state the reality conditions separately for these three forms. The following 
conditions refer to vectors, etc., in neutral space-time. 



86 Wave-tensor Calculus [6-5 

(1) Space vectors. 

The condition is that the electrical part, involving the matrices 



is antithetic to the kinematical part; or T e is antithetic to T k . 

(2) Vector densities. 

Since the vector density X-iTE &9 the above condition becomes 

(ZE 6 ) e is antithetic to (3^ 5 ) fc , 
so that the part of X, involving the matrices 

^16> ^159 ^25> -36> ^45> -5> 

is antithetic to the rest of the expression. 

(3) Strain vectors. 

Since the strain vector or three-dimensional density 8 = iTE^ 9 the 
condition becomes (SE ^ is antithetic to (8E^) k . 

By (4-66) the part of 8 which corresponds to the electrical part of I 7 is that 

containing v v v v v v 

o /& 23 , /& 31 , ^ 12 , /& 4 , ^ 5 , /& 46 , 

i.e. the real matrices. Thus the part of 8 containing real matrices is anti- 
thetic to the part containing imaginary matrices. We have therefore the 
simple result that the coefficients s^ of a strain vector are homothetic. 

(4) Kinematical rotations. 

These are restricted to ten components and their reality conditions, given 
in full in (6-23), secure that the part containing 

-15> ^25 > -^35 > -^45 

is antithetic to the rest. Comparing with the above results, we see that the 
matrix of a rotation must be regarded as a vector density. 

These conditions are founded on the result obtained in 6-3, that for a 
position vector JS? 5 a? 5 is antithetic to Ex+ Z? 2 # 2 4-^3^3 + jE? 4 # 4 . By 5-8, a 
position vector is part of a complete vector density. We have therefore to 
choose right- and left-handed frames related in such a way that the reality 
condition (2) for a vector density agrees with this. This justifies the choice 
made at the beginning of 6-4 when we took the connection to be that given 
by (3-93). 

The foregoing are the natural reality conditions in neutral space. But it 
must be understood that, since the only test is the invariance of physical 
reality, the conditions become less stringent as we limit the variety of 
transformations contemplated. Thus if we have a 5-vector U and a 10-vector 
F, forming a complete space vector U + V which satisfies the foregoing 
reality conditions, the combination U -f iV will violate them. But so long as 



6-5] Reality Conditions 87 

we confine ourselves to the ten kinematical rotations which transform 11 
and V separately, there is no special reason for preferring the combination 
U + VtoU + iV. It may well happen that U + iV is the customary form of 
combination. All we can say is that this latitude exists, because the expres- 
sion is a mental association rather than a genuine combination. Combination 
implies some loss of independence of U and F; so that a more general 
transformation is contemplated in which the combination U -f V persists 
and the combination U + iV is broken up. 

The hamiltonian H = E 1 p l + E 2 p 2 + E 3 p 3 + E^ m adopted in Chapter v 
satisfies the reality conditions for a vector density, since m is the com- 
ponent E^im. Accordingly J = ^* is a vector density. If we require the 
wave equation for a vector we must identify (p { , p 2 , # 3 , jp 4 ) with tho adjoint 
vector (J1&9J2&9 ./as > ^45) > an d therefore write the hamiltonian as 

H = E l5 p l + E 25 p 2 + Etopz + # 45 p 4 - m. (6-51) 

This satisfies the conditions (1) for a space vector. We need not stop to 
decide which of these forms is the most advantageous, because we shall find 
later that the form chiefly required in practice is that which corresponds 
to a strain vector. 

6*6. Charge and Spin. 

We have found in (5-83) that the complete stream vector of an elementary 
particle is, if the axes are suitably chosen, of the form 

J a = (E l + E 23 + EU + E IB ) im. (6-61) 

We have set a. = im in anticipation of the identification found below. We 
first notice that, since E 2S , E^ are real and E l , E IB are imaginary, J a satisfies 
the reality conditions for a space vector in neutral space-time ( 6-5 (1)). 

It may seem surprising that the result obtained in 5-8 is already adapted 
to neutral space-time, which was not formally introduced until later. The 
reason is that in the course of the derivation we selected the combinations 
of suffixes in accordance with experience-, and since we have no experience of 
electrically saturated space-time it was implicitly excluded. For a pure 
wave tensor in electrically saturated space-time the appropriate form is 

(E l + E 24t + E 35 + E 1B ) im, 

which is monothetic. It is easily seen that the momentum vector is then in 
the space-like direction E 3 . 
The hamiltonian is given by the last two terms of (6*61): 

H a = Eftim - E^im = E 46 ip + m. (6-62) 

We thus verify that m is the proper mass. The terms JE? 45 and E 19 in (6*61) 
accordingly represent the energy and the proper energy, which, although 



88 Wave-tensor Calculus [6-6 

equal in magnitude (owing to our special choice of coordinates), are exhibited 
separately in the wave tensor. 

To obtain a particle of opposite sign to J a we must reverse the sign of the 
electrical matrices, obtaining 

J b = ( E + #23 + #45 EM) im. 

The corresponding hamiltonian is H b - E^ip Q - m. It will be seen that if the 
standard form of the hamiltonian is taken to be 



the identification of m with the proper mass must be qualified by the 
proviso that this refers to the magnitude, and that the sign of m is that of 
the charge. The identification was based on (524) which gives m 2 , not m. 

The terms E l and U 23 describe a spin, the one giving the axis of the spin 
and the other the plane of the spin. But they play somewhat different parts, 
because one is reversed when we change the sign of the charge and the other 
is not. Clearly J5 23 , which is not reversed, represents the mechanical spin; 
and TJj , which is reversed, represents the magnetic moment. 

To obtain a particle of opposite spin we reverse the sign of J5 23 . In order 
that the wave tensor may remain pure, we must also reverse the sign of 
another term (5-5). We cannot reverse 7? 16 , since that would reverse the 
charge; we cannot reverse jE 45 , since that would give negative energy p (} , 
representing a "minus-particle". Hence we must reverse E. This confirms 
our interpretation of E l as a magnetic moment, whose sign depends on the 
direction of spin. 

Assuming arbitrarily that J a represents a positive charge with a "posi- 
tive " direction of spin in the plane x 2 x$ , we have the following classification : 

J a = im (E -f- #23 + ^45 + EM) positive charge, positive spin, 
J b = im ( - EL + EM + #45 ^ie) negative charge, positive spin, 
J c = im ( - E! - 23 + EM + E 16 ) positive charge, negative spin, 

J d = im (#JL - U 23 -f- E^ - E IB ) negative charge, negative spin. 

(6-63) 

The three-dimensional densities or strain vectors are also of importance. 
They are given by S = iJE^ . We obtain 

S a = im (E l -f #23 + EM -h E^ positive charge, positive spin, 
S b = - im (E 1 ^ 23 ,# 45 + # 16 ) negative charge, positive spin, 
S c = - im ( - # x # 23 4- JS/45 -I- E 1Q ) positive charge, negative spin, 

S d = - im ( E l + JE? 2 3 #45 4- E ie ) negative charge, negative spin. 

(6-64) 



6- 6] Reality Conditions 89 

We notice that in the strain vector the mechanical spin is represented by E 
and the magnetic moment by E^ opposite to the representation in the 
space vector. 

Since 8 a + S b + S c + S d = 4m, (6-65) 

the four forms can be regarded as resulting from a spectral analysis (5-7) 
of algebraic numbers. The properties of matter which is neutral as regards 
charge and spin can be represented by simple algebraic wave functions of 
the type introduced by Schrodinger. To introduce charge and spin Schro- 
dinger 's algebraic wave tensors are analysed into components $ a , S b , 8 C , S d , 
which can then be assigned modified probabilities independently, so that 
they no longer balance. This is the simplest way of connecting the theory of 
protons and electrons with the ordinary relativistic mechanics of neutral 
matter, as will be shown in 13-6. 

An entity represented by i (J a + Jb + J c + J d) or b Y I (8 a + 8b + S o + S d) 
will be called a neutral particle. The term is not intended to have any con- 
nection with the neutron. A neutral particle is not a combination of four 
particles; it is a single particle which (so far as our information goes) has 
equal probability of being a proton or electron and equal probability of 
either direction of spin. We regard S a , 8 b , fl c , 8 d as four elementary state* 
of a particle. In general the probabilities of the four states will be different. 
As a particular case (which, however, is the commonest case in practice) the 
probability of one of the states, say 8 a , may be unity; the particle is then 
classed definitely as a proton of positive spin. 

From the ordinary standpoint a neutral particle is a mathematical fiction, 
having no counterpart in experimental physics. Usually, if an experimenter 
knows anything at all about a particle, he knows whether it is a proton or 
electron. Direction of spin is less easily recognised; and a combination 
S + AS C , which gives a particle neutral as regards spin, may often represent 
the experimental knowledge available. 

The complete stream vector (space vector) of a neutral particle consists 
of a single component #45*45, and its three-dimensional density (strain 
vector) consists of a quarterspur E lB s^. Thus we have a very simple way of 
passing from electrical to neutral particles, namely by taking the quarter- 
spur of the strain vector an operation which corresponds to contraction of 
the corresponding wave tensor. 

In more general coordinates the term / 23 in the strain vector is replaced 
by three components jB 23 , E VL9 E 1Z . We have seen that these represent the 
magnetic moment. The electric moment (if any) will be represented by 
terms JE7 14 , E 24 , J 34 , since by the usual electromagnetic equations the 
magnetic and electric moments form a 6-vector. No such terms occur if the 
particle is at rest in the coordinate system; this is what we should expect, 



90 Wave-tensor Calculus [6-6 

since an electric moment implies a doublet, and cannot be associated with a 
single particle at rest. Electric moment terms are introduced if we set the 
particle in motion by applying a Lorentz transformation with matrix JE? 24 
or JE7 34 to (6-63). The electric moment thus produced is real. Dirac's theory 
seems to differ from ours on this point, since he obtains an imaginary electric 
moment for the electron.! 

6-7. Minus -particles. 

The wave tensor J a must presumably be regarded as expressing the 
absence of a particle of the kind represented by J a , or else an entity which is 
observationally equivalent to such absence. We may accept the current 
view (due to Dirac) that positrons and negatrons are minus-particles of this 
kind equivalent to the absence of electrons and protons. Thus 



represents a negatron. The sign of the last term shows that the charge is 
negative. 

The sign of the third term should signify that the energy is negative, 
according to the somewhat confusing definition of energy in wave mechanics. 
But the energy of a negatron or positron, as ordinarily understood, is cer- 
tainly positive. It is necessary to clear up this discrepancy of definition. 

Energy, momentum and spin are familiar conceptions in classical 
mechanics, and a complete energy -momentum -spin vector can be defined 
for a macroscopic system. Let us suppose that, by adapting this definition, 
we can assign to an elementary particle a complete vector T representing 
its energy, momentum and spin "as ordinarily understood". In wave 
mechanics we associate with the elementary particle a wave tensor J a = ^#*, 
whose components have at least some analogy with the energy, momentum 
and spin of classical mechanics. It is tempting to assume that T J a . But 
J n has the idempotent property that, apart from a numerical factor 
depending on the choice of units, J a 2 = Ja* We have therefore just as good 
reason to make the identification T = J a 2 . 

The proton and negatron which have opposite stream vectors J a , J a9 
have the same vector / a 2 . The identification T 7 tt 2 will accordingly make 
the energy of negatrons (and positrons) positive, as it should be. 

We therefore regard the primary wave tensor J = ^x* of an elementary 
particle as a charge-current vector, and 7 2 as the true energy-momentum 
vector. We have been considering the vector J for simplicity, but strictly 
the relation is between the three-dimensional densities S, $ 2 ; the charge- 
current density is represented by S, and the energy-momentum density 
byfl 2 . 

t Quantum Mechanics, 2nd ed., p. 263. 



6-7] Reality Conditions 91 

Since /S a is an algebraic multiple of /S, the two vectors coalesce for an 
elementary particle. This has happened by design rather than by accident. 
The coalescence of S* with S is virtually the definition of an elementary 
particle; for elementary character has been taken to correspond to purity, 
and purity to idempotency. This helps us to understand why so much im- 
portance is attached in wave mechanics to resolution into pure or idem- 
potent constituents. It is for these constituents that we are able to replace 
a quadratic dynamical property by a linear property. For impure con- 
stituents these would be distinct wave tensors describing different pro- 
perties. Broadly speaking S 2 represents mechanical characteristics and 8 
electrical characteristics; they become unified only in an elementary particle 
whose momentum and current coincide. 

This agrees with general relativity, in which the mechanical properties 
are specified by an energy tensor of the second rank (a quadratic function 
of the individual velocities), and the electrical properties by a charge- 
current vector (a linear function of the individual velocities). 



CHAPTER VII 
STRAIN VECTORS AND PHASE SPACE 

7- 1 . Internal Wave Functions . 

In classical mechanics it is usual to resolve the motion of a system of par- 
ticles into a motion of the centre of mass, and a motion of the individual 
particles relative to the centre of mass. We shall distinguish these as the 
external and the internal motions of the system. 

Similarly in wave mechanics we resolve the motion of a system into an 
external motion specified by external wave functions, and an internal motion 
specified by internal wave functions. So far as the external motion is con- 
cerned, the system is equivalent to a single particle located at the centre of 
mass. It is characterised dynamically by an external momentum vector 
(lh 9 Pz > #3 > jPo) an( i a proper mass m. If the space-time frame of reference for 
the external motion is changed, the momentum vector undergoes rotations 
and Lorentz transformations. The theory of the external wave function 
coincides with that of a simple particle. 

The internal wave function introduces new ideas. Lorentz transformations 
are not applicable to the internal motion; for, by definition, the internal 
motion is relative to the centre of mass, and, if we applied a Lorentz trans- 
formation to it, it would be referred to some other standard of rest. The 
time-axis of the frame of reference for internal motions and wave functions 
must agree with the direction of the external momentum vector. 

Thus we have a uniquely defined space-time frame for internal motions, 
and "simultaneity" has a definite meaning provided that the system is 
not so extensive as to make it necessary to take account of curvature of 
space-time. It is part of our mental conception of a complex system that 
it is a simultaneous aspect of its several parts. Each particle has three 
internal (relative) coordinates ^ = 2^ x^ and the momenta conjugate to 
f^ are the internal momenta of the particles of the system. There is one 
time-coordinate s common to the whole system. If the same coordinate 
system is used for the internal and external motions, the external time will 
also be s. But in general different frames will be used, since it would be idle 
to consider external motion if the external frame had always to be chosen 
so that the system was at rest in it; the external time t will then differ from 
the proper time s. 

The independent variable for the internal motions and wave functions is 
always the proper time s. It should be noticed that if, for special purposes, 
separate internal time-coordinates analogous to the internal space co- 
ordinates are assigned to the particles, these will be T=$ t. To associate t 



?!] Strain Vectors and Phase Space 93 

with Xp - Xp (as is sometimes attempted) is a hybrid procedure unwarranted 
by any theoretical principle. 

For internal wave functions Lorentz transformations are definitely ruled 
out; but relativity rotations in three-dimensional space are applicable. The 
direction of the time-axis is prescribed by the external momentum vector; 
but there is no corresponding specification of the orientation of the other 
axes. 

The use of internal wave functions which, by their very nature, cannot be 
subjected to Lorentz transformations is often called "non-relativistic 
treatment" with the implication that it conflicts with the principle of 
relativity. This is a misunderstanding of the nature of the Lorentz trans- 
formation and its place in relativity theory. So long as we deal with quan- 
tities defined as relations between physical entities relative coordinates, 
relative velocities, relative momenta we are on safe ground. (By relative 
coordinates we do not, of course, mean the difference of coordinates in some 
arbitrary frame of reference,! but the coordinates of one particle in the frame 
in which the other particle is at rest at the origin.) It is when we introduce 
into our formulae analogous quantities relative, not to physical reference 
objects, but to abstract frames of space and time, that Lorentz invariance 
is demanded. For we are then unable to define which of the equivalent 
frames has been selected to which frame our formulae apply so that, if 
the formulae are to mean anything at all, they must have a form invariant 
for all transformations of the frames. 

In relativity theory itself there has been no such tendency to let Lorentz 
invariance grow into an obsession. One of the best known formulae in 
relativity theory is ds 2 = -y- 1 dr 2 -r 2 d0 2 -r 2 sin 2 0d^ 2 H-ydi 2 for the line 
element in the gravitational field of a particle. This is not invariant for 
Lorentz transformations; but we can scarcely describe the formula which 
is the soiirce of the three crucial tests of Einstein's theory as "non-rela- 
tivistic". If Lorentz invariance is not demanded in the investigation of 
the motion of a planet round the sun, it can scarcely be demanded in the 
investigation of the motion of an electron round a nucleus. 

Hitherto, in developing the analytical theory, we have had in mind the 
motion of a particle or the external motion of a system. In the present 
chapter we shall introduce the modifications appropriate for treating the 
internal configuration and motion of a system. There are two fundamental 
differences. Firstly, the time reckoning will now be a proper time, fixed by 
the external momentum vector, and therefore invariant for any permissible 
transformations of the variables describing the internal configuration. 
Secondly, whereas change of the coordinates x^ of the centre of mass is a 

t This possible confusion of meaning does not arise in the case of relative momentum, 
which obviously has no connection with difference of momentum. 



94 Wave-tensor Calculus [7-1 

relativistic change, all points in the space-time hypersphere being equi- 
valent, change of the relative coordinates f^ is in general a strain or intrinsic 
distortion of the system, and is therefore not to be represented as a relativity 
rotation of the vectors describing the internal state. 

7- 2 . Co variant Wave Tensors . 

We shall now consider the transformations of a covariant wave tensor of 
the second rank 8^. Like the mixed tensor it is a matrix composed of 16 
elements; and it may be resolved into matrix components in an orthogonal 
frame by the same formula 8-I^s^E^ but s^ will not be a space vector. 
We shall call s^ a strain vector. Thus a strain vector corresponds to a co- 
variant wave tensor in the same way that a complete space vector corre- 
sponds to a mixed wave tensor.f 

The transformation formula for a covariant wave tensor has been found 
in (1-53), namely S' = qSq. (7-21) 

Let q = e^ E f e . First let E^ be an antisymmetricai matrix, so that 



Then (**" V = cos W . ( 1 ) ]3a + sin *0 . 



That is to say, 9^9"^ 

If EH is a symmetrical matrix, we have q = q. Hence (7-21) becomes 
S' = qSq~ l for antisymmetricai transformation matrices! 
= qSq for symmetrical transformation matrices. } 

The transformation qSq~* agrees with that of a mixed wave tensor (1-463). 
The transformation qSq has been called an antiperpendicular rotation 
(4-16). We have therefore the result: 

A strain vector behaves as a space vector when the transformation matrix is 
antisymmetricai, but undergoes antiperpendicular rotations instead of the 
corresponding ordinary rotations when the transformation matrix is sym- 
metrical. 

As in the case of space vectors, we shall use the name "strain vector" 
indifferently for the array of components s^ or for their symbolic combina- 
tion S^XSpEp. 

A strain vector may be represented graphically by plotting its com- 
ponents s in a 16-dimensional space. But it must be remembered that the 
line which represents it will be thought of as a space vector. When a tensor 

t This is the fundamental definition of a strain vector. It will be shown in 7-6 that 
three-dimensional vector densities are strain vectors, and they have therefore been called 
strain vectors in anticipation ( 4-6). 



7-2] Strain Vectors and Phase Space 95 

transformation is applied, all geometrical reference lines in the space rotate 
as space vectors; the strain vector follows its own transformation law, and 
is displaced relative to this background of space vectors. We shall now 
examine this relative displacement. 

Let S be a strain vector and T a space vector, and at first let T coincide 
with 8. Perform the infinitesimal transformation g = e* de , where d is a 
general infinitesimal matrix. We write d*=d 8 +d a , where d 8 , d a are 
its symmetrical and antisymmetrical parts. Since d is infinitesimal, q is 
equivalent to the transformations </ a = e* d and g a = e^* applied succes- 
sively. Thus we have 

S'-Matffc- 1 *., T f ^q s q a Tq a ^q 8 -\ (7-23) 

And, since 8 = T, S' = T'q* = T'e d *. (7-24) 

Hence, d 8 being infinitesimal, 

S'-T' = T'd 8 = Sd 8 . (7-25) 

We take S to be non-singular. Then 8d 8 ^0 for any non-zero value of 
d 3 . Since the general symmetrical matrix contains ten independent con- 
stants, the relative displacements Sd 8 will occupy a ten-dimensional space. 
We call this the phase space of the strain vector. 

To exhibit this graphically, let S and T be represented by the lines OQ 9 
OP in the 16-space. A tensor transformation of the space vector is repre- 
sented by keeping OP fixed and rotating the axes of reference, thereby 
altering the components ^ referred to the axes; thus P can be regarded as 
a fixed origin. At first Q coincides with P; but the transformation produces 
the relative displacement PQ = 8' T'. The interesting point is that, 
although there are 16 independent tensor transformations, Q is limited to 
a ten-dimensional locus. Provided that 8 is non-singular, every direction 
in this locus is a possible direction of PQ. 

This construction gives only an infinitesimal region of phase space, and 
we must extend it by a process of continuation. The problems which arise 
in constructing the complete phase space will be considered later. 

We have seen that antiperpendicular rotations represent intrinsic 
deformations of the physical system considered. The points of phase space 
therefore represent in systematic order different intrinsic states or con- 
figurations! of a system described by a strain vector. This corresponds to 
the usual definition of a phase space in statistical mechanics. In the applica- 
tions for which it is intended, phase space is occupied by a probability 
distribution. 

t I shall generally use the term configuration; it is to be understood in its broadest sense. 
State would express the meaning better; but I was anxious to avoid using a term which has 
been given a technical significance by Dirac. The term phase is often used in statistical 
theory; but I have reserved it for the angular variables occurring in the exponentials. 



96 Wave-tensor Calculus [7-3 

7*3. Real Phase Space. 

For treating strain vectors and phase space we adopt a frame E^ consisting 
of four-point matrices. The E^ then consist often symmetrical and six anti- 
symmetrical matrices; the symmetrical matrices are imaginary and the 
antisymmetrical matrices real. In 6-1 space-like directions have been 
associated with imaginary matrices and time-like directions with real 
matrices; it is convenient to use these descriptions generally as a nomen- 
clature rather than as an anticipation of the physical manifestations of the 
vector components associated with them. Having regard to a future 
extension to double wave tensors, the definition is best stated in the form: 

A space-like matrix is homothetic with its eigenvalue; a time-like matrix is 
antithetic to its eigenvalue. 

The eigenvalues of the E^ are imaginary. Thus, for simple wave tensors, 
we have the equivalence: 

Space-like = symmetrical = imaginary matrices, 
Time-like = antisymmetrical = real matrices. 

The ten dimensions of phase space are space-like. 

We can no longer represent the E^ by general fourfold matrices, since 
these are usually neither symmetrical nor antisymmetrical. At first sight 
the limitation to a particular frame of matrices seems a serious loss of 
4 * relativity". But the application of this chapter is to the internal wave 
tensors of a system, which, as we have seen, are not subject to Lorentz 
transformations. It is not merely permissible to use a fixed frame; it is 
essential that the frame to which these internal tensors are referred should 
have some quality which resists Lorentz transformations. J 

In treating external space we considered it to be a drawback that in 
matrix representation the E^ have properties additional to those which 
they are defined to have as constituents of a complete orthogonal set. But 
the additional properties come in useful in internal space, because we have 
to indicate a distinctive direction, viz. that of the external momentum 
vector, and the planes of simultaneity orthogonal to it. We do this by giving 
to the matrix J5 4 , belonging to the distinctive direction, the property of 
antisymmetry; matrices belonging to directions in the plane of simultaneity 
are symmetrical; matrices belonging to intermediate directions are neither 
symmetrical nor antisymmetrical. 

We have seen that one of the difficulties of applying to physics an algebra 
comprising complex numbers is that half the mathematical possibilities 
have to be set aside as unreal, i.e. not corresponding to actual phenomena. 
We approach the problem of determining the reality conditions of phase 

f It is the Lorentz transformations of the frame which introduce asymmetry. For spatial 
rotations the matrices preserve their symmetry or antisymmetry. 



7-3] Strain Vectors and Phase Space 97 

space somewhat differently from the similar problem in Chapter vi. The 
feature of phase space is that it is the seat of a probability distribution. If the 
probability distribution does not extend to the unreal configurations if they 
have zero probability there is no need to stigmatise them further. Accord- 
ingly the discrimination of the world of real phenomena from the world of 
unreal phenomena is made when we insert the probability distribution. We 
shall show that this fixes the reality conditions of phase space unambiguously . 

The method of statistical mechanics contemplates an initial or a priwi 
probability distributed over phase space so that the probability in any 
volume is proportional to the volume. Observational information modifies 
this initial probability, so that the actual pro bability concerned in a particular 
problem is the product of the initial probability and a modifying factor. It 
is essential to the method that the phase space over which the probability 
is distributed should have finite volume; for if it were infinite, the initial 
probability associated with any finite region would be zero, and the method 
would break down. This requires that phase space should be a closed space. 

It is true that statistical mechanics is often applied to a space closed by 
a supernatural barrier instead of by its own re-entrancy . But the barrier is 
merely a compression of the curvature required to close the space into a 
singularity. If we reverse the motion of the particles by natural fields of 
force instead of supernaturally, space must be curved to represent these 
fields. No ordinary field is such that an electron has zero probability of 
leaking through; to confine it rigorously the curvature must be sufficient to 
close the space. 

Thus, in order that our phase space may be the seat of a probability dis- 
tribution, it must be closed. This means that the matrix dQ^^E^dQ^, 
which gives the displacements in phase space, must correspond to circular, 
not hyperbolic rotations. This requires that the dO^ shall be real (cf. (4-15) 
and (6-13)). Then, since the symmetrical E^ are imaginary, d& 8 is imaginary. 
We have the result: 

In order that the displacement in phase space given by the transformation 
q e \d s ma y represent a real change of configuration, d 8 must be an 
imaginary matrix. (7'31) 

This reality condition may also be expressed by saying that the space is 
a phase space in the other sense of the term phase. Instead of a single algebraic 
phase 8 indicated by a factor e ie , we have ten phases for which i is replaced 
by different matrix roots of - 1. Since the 0^ are real, they are real periodic 
phase angles in the ordinary sense. 

We can see this most easily by taking the initial value of the strain vector 
to be unity. This involves no loss of generality. By (7-24) 8' = T'e d **. Since 
S' and T' are non-singular, T' has a reciprocal T'" 1 ; then 



98 Wave-tensor Calculus [7-3 

The transformation law of T-*S is 



so that T~ 1 S is a strain vector, and its initial value is 1. Thus the phase space 
generated by the transformations of an arbitrary non-singular strain vector 
8 may equally be regarded as generated by the transformations of a unit 
strain vector, viz. (1)' = 1 e d& *. (7'32) 

If d& 8 does not involve non-commuting components, (7-32) can be in- 
tegrated so as to apply to finite displacements. Thus there will be a line of 
configurations for which the strain vector is 



If is real, the strain vector repeats itself at intervals 0^ = 2?r, so that we 
return to the original configuration. In other words phase space is re-entrant 
in the 0^ direction. If 0^ is imaginary, the strain vector is non-periodic and 
the phase space is open in that direction. Hence, in order that phase space 
may be closed, real configurations must correspond to real phase angles 0^ . 

Following an arbitrary track in phase space the transformations are in 
general non-commutative. Thus the exponentials combine by non-commuta- 
tive multiplication, and the increments of the phases combine by non- 
commutative addition. The ten-fold phase is therefore non-integrable; this 
means that it must be represented in a curved space. 

By 3-6 the transformations g = e* A V^^, in which dO^ is real, are all 
unitary. Thus a strain vector which is non-singular initially remains non- 
singular throughout phase space. 

By (7-31) and 6-5 (3) the matrix d& s , which represents displacement in 
phase space, satisfies the reality conditions for a strain vector, but not those 
for a space vector or vector density. We must therefore take d 8 to be a 
strain vector. 

7-4. Coordinates in Phase Space. 

Let g = e id be the transformation which displaces a point Q to a neigh- 
bouring point Q' in phase space, and let 

d&^^E^. (7-41) 

Then the ten 0^ provide locally a coordinate system for describing points Q' 
near an origin Q. We call these local orthogonal coordinates or natural 
coordinates. A corresponding system of linear coordinates x^BO^ is 
introduced by attributing to the phase space a definite scale constant jR. 
For the present B must be considered arbitrary, since it could only be 
defined by introducing relations to an extraneous system. 
The volume of a ten-dimensional element of phase space is defined to be 

do>=d0 1 d0 2 ...d0 10 . (7-421) 



7-4] Strain Vectors and Phase Space 99 

It is a scalar quantity. We can, if we prefer, write it as E l dd l .E 2 dd 2 ... E w dd w , 
but the product of the ten matrices is found to be 1. We thus have a 
definition of equal volumes at different points of phase space, each volume 
being measured in terms of natural angular coordinates at the point where 
it lies. By this definition equivalent volumes are equal volumes. As in 2*9 
equivalent volumes are formed by making the same construction in 
equivalent frames. 

For any other system of coordinates x^ , we have in the notation of general 

relativit y d^V^g.dr, (7-422) 

where dr = dx^dx^ . . . dx lQ . 

In particular for natural linear coordinates 

V^7==jR- 10 . (7-43) 

The local orthogonal system is only applicable when the squares of 0^ are 
neglected. The most important equations of physics are differential equations 
of the second order, and in order to investigate them it is necessary to intro- 
duce a coordinate system valid at least as far as the squares of the co- 
ordinates. This problem will be treated in the next section. We have not 
much occasion to employ the properties of phase space as a whole, and our 
methods are chiefly adapted for treating an infinitesimal region. But it is 
important for our theory to prove that the whole volume of phase space is 
finite. Although this seems rather obvious, I have had some difficulty in 
proving it formally. We have adopted circular rotations in order to secure 
finiteness; but until we examine how they are to be extended beyond an 
infinitesimal region we cannot be sure that they will achieve this end. They 
secure re-entrancy along geodesic tracks, but we have still to prove that 
phase space has no tortuous exit. Although phase space has the same kind of 
uniformity as a hypersphere, it is different from a hypersphere; it contains 
pairs of antiperpendicular directions. Thus ordinary spherical coordinates 
do not apply, and I do not know of any suitable adaptation of them. 

There is a fundamental difficulty in specifying finite deformations or 
strains of a system, which arises in the following way. Let A l9 ^4 2 denote 
two different orientations of the same (unstrained) system, and let A 1 
denote the system in a strained condition; how are we to decide whether the 
strain should be measured by A' A l or A' A 2 1 There is no absolute 
one-to-one correspondence of the orientations of strained and unstrained 
systems; but it is necessary to lay down some conventional rule which will 
prevent our representing the same deformation twice over as A' A l and 
A'-A 2 . 

We have secured a unique representation of infinitesimal strains. The 
initial (unstrained) state was represented by S or T\ the strained state 8' 



100 Wave-tensor Calculus [7-4 

was then compared with T", which represents the unstrained state in a 
different orientation connected with S' by tensor rules. This cannot be 
extended to finite regions, because the transformations T-+T' are not 
integrable. To meet this difficulty, consider the transformation 

3 =n^ = IIe*^M, (7-44) 

the product consisting of ten factors with space-like E^ arranged in a fixed 
order. The elementary transformations q^ are applied successively (in the 
reverse order to that in which they are written). The ^ are not infinitesimal. 
Applying this transformation to a strain vector, so that $->$', we obtain a 
strained configuration which will correspond to some point Q f in phase space. 
We can adopt ^ as the coordinates of Q'. 

Let one of the ^, say <f> a , receive an increment d<f> a . Let the new point 
be #", and the new value of q be q". For two matrices q, q" which differ 
infinitesimally we can find a matrix d such that 

q" = e***q. 
It is easily seen that d = XE d<j> X-*, (7-45) 

where X is the part of the product IT which precedes q a . X cannot be singular 
(3-6). In general d will include time-like matrices. Since e* rfw is the 
transformation which changes Q' to Q", the components of d s are the 
natural coordinates dO^ of Q" referred to the origin Q'. 

Similarly, we can express the other displacements d^ in natural co- 
ordinates dOp at Q'. The jacobian 3 (^J/3 (0^) gives the ratio of the volume 
element dr = dfadfa . . . d<f> w to the natural volume element da) = d9 l d6 2 . . . d0 10 
and hence determines V g in the usual formula da) = V gdr. 

The half-period of each of the ^ is 2?r. After each half-period the values 
of q repeat themselves with opposite sign, and the corresponding strain 
vectors repeat themselves; so that the half-period represents a circuit of 
phase space. Hence for the whole of phase space, or of that part of phase 
space covered by the coordinate system ^ , 



Thus unless V g becomes infinite anywhere, or unless there are configura- 
tions not obtainable by the transformation (7-44), the volume Jdeo is finite. 
Since X cannot be singular, (745) shows that 30^/3^, is not infinite, and 
hence V~-^g is never infinite. There are, however, loci where V-gr = 0. For 
example, if the first two factors of II are e*^"e*^, where E v , E a anti- 
commute, we have by (7-45) for a displacement d<f> a 

d = e^^E a d^ a e^^ e E ***E a d<t> . 

On the locus ^ p = i^, this becomes E v E a d<f> a . The product of two anti- 
commuting space-like matrices is a time-like matrix; thus d is wholly 



7-5] Strain Vectors and Phase Space 101 

time-like, and d 8 vanishes.f But this type of singularity, in which a small 
natural volume is infinitely magnified in coordinate volume, is harmless for 
our purpose. 

I have little doubt that a geometer could furnish a more elegant proof. 
Probably he could evaluate the volume. But 1 think the above investigation 
satisfies us that the volume of phase space is finite. 

7*5. Stereographic Coordinates. 

One of the most important practical steps in the theory is to provide the 
analytical machinery for investigating differential equations of the second 
order. For this purpose we introduce a system of coordinates valid (under 
certain restrictions) for finite regions of phase space. 

On a hypersphere of radius R, Stereographic coordinates are such that the 
line element is 



_ )? (7 . 51) 

where r a = # 2 + y z + z 2 + . . . . The coordinates are thus locally orthogonal and 
isotropic, but not uniform; the actual length ids is A r times the Euclidean 
length (dx 2 + dy 2 +dz 2 +...)*, where 

A^l + r 2 /^ 2 )- 1 . (7-52) 

The hypersphere is in this way projected into a Euclidean space with a 
variable gauge factor A r . 

Analogous coordinates x^ in phase space are defined as follows. Let 
X = S 8 ^o; /t , the summation being restricted to the space-like matrices. 
Then Stereographic coordinates are such that the displacement from the 
origin to the point x^ corresponds to the transformation 

The right-hand side is to be interpreted by expanding in infinite series. This 
formula is limited to a domain containing only perpendicular coordinates 
together with its infinitesimal neighbourhood in all directions. That is to 
say, X is limited to a pentadic expression or to a single algebraic variable, 
but dX is unrestricted. The inclusion of the infinitesimal neighbourhood is 
essential, because these coordinates are used principally when we are treating 
the complete ten-dimensional volume element of phase space. 
Accordingly X* is algebraic, J and we set 

Z 2 = -r 2 . (7-54) 

t This is illustrated graphically by taking E ff , E to correspond to rotations of a sphere 
in the planes xy, yz. Taking an origin on the x axis, a displacement from the origin to any 
point on the sphere can be represented by two such rotations in the order given. Different 
values of <f> a , <f> v will give different points, unless </> v = k 7r or <t><j~^- This example shows 
that the singularity of the representation at these two points does not signify a failure of 
the coordinate system ^ M to cover the whole of the space. 

% It is tempting to describe the domain of X as being limited by the condition that X 2 is 
algebraic; but we define it more stringently in order to exclude compact ^-numbers (5*66). 



102 Wave-tensor Calculus [7-5 

If displacement from x^ to x^ + dx^ corresponds to the transformation 
e** e , we have by (7-53) 

<* JHJW*_ (l(1Xm\ l . (7-55) 

C tl-Z/2JZJ -\l-(X+dX)l2R\ l j 

When -2C is limited as above, the most general matrix dX can be divided into 
two parts, one of which commutes and the other anticommutes with X\ for 
the limitation secures that a complete set can be formed with X/r as one of 
its members; and if dX is resolved in that frame, its components either 
commute or anticommute with X/r. The transformations corresponding to 
the two parts of dX can be treated successively, since they are infinitesimal. 
First, let dX commute with X. Then, since there are no non-commutative 
symbols, (7-55) can be solved like an algebraic equation, giving 



d@ 



| log r 



Next, let dX anticommute with X. Then the differential of any even 
power of X is zero. Since 



dX/2R _ 



Hence by (7-55) 



By definition Jf and dX contain only space-like matrices, and since they 
anticommute their product is time-like. Thus (7-56) and (7-57) give the 
same value of d 8 d ^ (l + r *i B *)-i dX IR = \ r dXIR (7-58) 

by (7-52). 

Thus the differentials dx^ are the natural linear coordinates at the point 
considered, but the scale constant B/\ r is variable precisely as in (7-51). 

The volume of a ten-dimensional element being 

dw = dO^dO^ . . . d9 lQ = V gdx^dx^ . . . dx lo , 
we have by (7-58) _ 

V - g = (A,/!?) 10 = J2- 10 ( 1 + r 2 /.R 2 )- 10 . (7-59) 

The following theorem is required later: 

If a transformation X (not infinitesimal, but not containing antiper- 
pendicular components) is applied to the strain vectors, the stereographic 
coordinates of all points in the infinitesimal neighbourhood of the origin are 
changed by the same amount, to the first order. 



7-6] Strain Vectors and Phase Space 103 

Let the stereographic coordinates 0, dx^ of the origin and a neighbouring 
point be changed by the transformation to x^ , x^ + dx^ ; and let dX = S E^ dx^ , 
etc. We have to show that dX'=dX. The transformation is that given by 
(7-53), and we have 

ilX!2R^ _ 
U-Z/2IZJ ~ 

where d3T" differs (if at all) from dX' by including time-like components,! 
By the conditions imposed on X , we can divide dX into two parts which 
respectively commute and anticommute with X, and treat them separately. 
For the commuting part the left-hand side can be treated algebraically, and 
gives immediately the required result dX" = dX. For the anticommuting 
part we proceed as in obtaining (7-57) and find that dX" differs from dX by 
time-like components only, so that dX' = dX. 

This result may also be stated in the form: In stereographic co- 
ordinates, a finite displacement of the above restricted type commutes 
with all infinitesimal displacements. 

The results of this section are used extensively in Chapter xn. 

7-6. Associated Strain Vectors and Space Vectors. 

We employ four-point matrices, jB 4 , E 6 being as usual the real members of 
a pentad. Let e^ be a covariant wave tensor which has the value JSL in the 
coordinate system initially chosen. Then, after a transformation g, we have 
fysssgJE^gr 1 or qE^q according as q contains a time-like or a space-like 
matrix (7-22). 

The wave tensor e 46 has the remarkable property that it is invariant for the 
ten rotations in five dimensions (kinematic rotations). This is easily verified, 
remembering that by the above formulae it is unaltered by transformations 
with real matrices with which it commutes or imaginary matrices with 
which it anticommutes. Thus for all orientations of the axes in five dimen- 
sions * 



No other strain vector has this property. It may be compared with the 
metrical tensor g^ = 8^" and the contravariant tensor density * V(rr in ordinary 
tensor calculus, which are likewise exceptional in having invariant values. 
In ordinary tensor calculus we define associated covariant and contra- 
variant vectors A^ A* by the relation Ap^g^A*. We shall now define 
associated (initial) covariant and contravariant wave vectors 0*, #*. A 
linear relation between them must be of the form 



t The left-side gives the transformation -> cfo M followed by the transformation -> x^ . 
By definition this takes us to the point # M +<fo/ . We cannot immediately identify it with the 
direct transformation -> x^+dx^', since an infinitesimal time-like matrix may be included. 



104 Wave-tensor Calculus [7-6 

where a^ is a covariant wave tensor. Consider the special case a^p = i (e 45 ) a . 
Then ^^t'x* (45)00 by (7-61); or, dropping suffixes, 

f = *y 45 . (7-62) 

Wave vectors connected by the relation (762) will be called associated wave 
vectors. 

Thus in wave-tensor calculus, (iE^)^ plays the part of g^p in the operation 
of lowering a suffix. f Since (i# 45 ) 2 =l, we have also x* = ^*^?45, which 
defines the operation of raising a suffix. Since four-point matrices are used, 
raising or lowering a suffix is a rather simple process; for example, in the 
standard pentad (3*27), 

J 45 = 0010 

0001 

-1000 

0-100 

Hence &,**,**, #4 = *' (* 3 > X 4 , " X\ ~ X 2 )- (7-63) 

Now multiply (7-62) by a covariant wave vector ^ by outer multiplication. 
We have * 



Or, denoting the covariant wave tensor 0<* by S, and the mixed wave 
tensor fc* by T, !3=iTE K . (7-64) 

We have therefore the important result: 

If a space vector T is multiplied by iJ5? 45 we obtain an associated strain vector 
S. Reciprocally, if a strain vector 8 is multiplied by iJE7 45 we obtain an associated 
space vector T. 

The components of S are obtained by shuffling the components of T, and 
in some cases inserting factors i\ the process is thus somewhat analogous 
to (7-63). We have already given the precise relation between the s^ and ^ 
in (4-66). 

The relation (7-64) is invariant for kinematical rotations, but electrical 
rotations ( 6-4) are excluded. It would be undesirable to exclude electrical 
rotations of a strain vector; because a strain vector is used primarily in 
connection with phase space, and some of the directions of displacement in 
phase space correspond to electrical matrices. But we have suggested that 
the term "space vector" implies that only kinematical rotations are con- 
templated ( 4'3). It would appear therefore that the strain vector is the 
more fundamental conception, and that space vectors are a derivative 
conception introduced into physics by the formula (7-64). That is to say, 
we take 8 to be a covariant wave tensor for all transformations; then the 

t If we could depend on the suffixes appearing explicitly in the formulae, we should 
naturally use the notation x", # a for the two associated wave vectors; but since suffixes are 
generally omitted, we have to distinguish them by different letters x> <f>. 



7-6] Strain Vectors and Phase Space 105 

matrix T determined from it by (7-64) will behave as a mixed wave tensor 
for kinematical rotations but not for electrical rotations; so that it will be 
rigorously a space vector, but only imperfectly a mixed wave tensor. If all 
space vectors in physics originate out of strain vectors in this way, we see 
why there is never any occasion to employ in practice the extra relativity 
transformations possessed by space vectors derived from mixed wave 
tensors. We have previously attributed the absence of these transformations 
to the neutrality of space-time. The two explanations are connected; because 
a system composed of particles of one sign could have no equilibrium con- 
figuration, and give no foothold for statistical mechanics with its attendant 
conception of phase space. 

Since (7-64) is the same equation as (4-65), S is the three-dimensional 
density of T\ and we reach our earlier definition of a strain vector as the 
three-dimensional density of a space vector. The relation is reciprocal, and 
T is also the three-dimensional density of 8. But the discussion in 4-6 was 
limited to an infinitesimal region round the origin, where the volume element 
of the 3-space was dW J2 ^ = iE^dw. If we move away from the origin, the 
matrix of dfF 123 will no longer be the original ,645. In addition to the change 
of direction of the normal to space-time, we must allow for an arbitrary 
rotation of the time direction, since there is no absolute way of defining the 
reckoning of simultaneity over an extended area. The general formula for 
the volume element is dW 123 =i(e u )J>dw, (7-65) 

where (e 45 ) a ^ is the space vector which has the value E^ in the original co- 
ordinate system at the origin. Hence the three-dimensional density of T is 
iT (^g)/, whereas the strain vector is iTE^^iT (e 45 ) aj3 . 

We see therefore that the elementary definition of a strain vector as the 
three-dimensional density of a space vector, and vice versa, does not hold for 
an extended curved region. It can, however, be preserved if we represent 
the element of volume as a strain vector with matrix (e 45 ) a instead of in the 
more familiar way as a space vector with matrix (e 45 ) a ^. To regularise this 
we distinguish an internal and an external three-dimensional space. They 
consist of the same points; a particle which is in one is in the other; but the 
metrical conceptions are different. The internal space is part of phase space, 
corresponding to three of its ten dimensions. We have seen (7-421) that the 
whole volume element is scalar; the separation of the three and the seven 
dimensions is so drawn that each combination is a strain vector with matrix 
JS7 45 , constant throughout phase space, and therefore in a sense characteristic 
of the whole phase space. 

The phase space, including the internal space, is described wholly by 
strain vectors. It will be remembered that the matrix d@ 8 , determining 
displacement in it, was found to be a strain vector ( 7-3). But when the 



106 Wave-tensor Calculus [7-7 

three dimensions are separated from the others, the densities or fluxes of 
strain vectors with respect to the three-dimensional volumes constitute 
space vectors. These require for their representation an external space. 

7*7. Normalised Strain Vectors. 

We shall now transform the wave equation so that it applies to strain vectors. 
The equations giving the factors iff, x* f a space vector are 

/fy = 0, X *ff = 0, (7-71) 

where H = E^Pi + E 26 p 2 + E 3b p 3 + E^p^ - m, by (6-51). Let 

H 8 = - EM H = Eu Pl + E 2 tp 2 + E 3 tp 3 +pt + E^m. (7-72) 
Then (7-71) becomes E^H 8 ^ = 0, x* E *& H 8 = > 
so that #0 = 0, <*# 8 = 0, (7-73) 

where <* = *x*^45 by (7-62). These are the wave equations for the factorisa- 
tion of a strain vector ^*. The hamiltonian H s is part of 0^*. If p^ ipo, 



we have . (7-74) 



This satisfies the reality conditions for a strain vector ( 6-5 (3)), the coeffi- 
cients in (7-74) all being real. The energy p Q is associated with the algebraic 
matrix J57 16 ; and the coordinate conjugate to it, namely the time t, must also 
be associated with E 1Q . Accordingly in phase space the algebraic phase 
represents the time. 

Since the phase space corresponds to the internal configurations of 
a system, the time 16 is measured in the direction defined by the external 
momentum vector ( 7-1). 

In phase space the algebraic phase may be separated from the other phases, 
leaving a nine-dimensional space. This is permissible, because the algebraic 
phase commutes with all the others. If dco c is the nine-dimensional volume 
element and d0 16 the algebraic phase, so that 

da> = daj c d6 u , (7-751) 

we can treat d0 16 separately in integrating; so that, if Q is the volume of the 
ten-dimensional phase space, and ti c the volume of the nine-dimensional 

s P ace ' a=Q c .27T. (7-752) 

In defining phase space we associated a strain vector 8 with each point. 
We shall now more definitely associate a strain vector S = Sdo> c /Q c with a 
range of configurations rfo> c . This strain vector will serve three purposes: 

(1) Its algebraic phase indicates the time. 

(2) Its symbolic phases describe a particular configuration. 

(3) Its amplitude indicates the probability of the system having this 
configuration to within a range do> c . 



7-7] Strain Vectors and Phase Space 107 

This interpretation applies differentially to small changes of , but it is 
not so easy to interpret S itself. For the latter purpose we must introduce 
the determinant of S. Since det S is invariant for all non-algebraic displace- 
ments in phase space ( 3-6) it is independent of (2). For a purely algebraic 
strain vector, detS = S 4 . If then we take as "origin" the configuration for 
which the strain vector is algebraic (regarding this as the standard un- 
strained condition), we have 

(1) The time is the argument of the complex number (det Z)*. 

(2) The configuration is specified by the matrix S/(detS)*. 

(3) The probability is the modulus | (det S) | . 

It is a most important feature of the symbolic theory that tlie same 
symbol specifies both the configuration and the probability that the system has 
that configuration. 

The probability distribution which we contemplate initially is uniform 
throughout phase space, so that an element dcu c contains probability dco c /l c . 
Hence for the initial probability | (det S)l \ = 1. In particular, at the origin 
(standard unstrained configuration) S=l. This initial or a priori distribution 
is the framework the "blank sheet" into which we insert whatever we 
may learn about the system by special observation. If the observational 
evidence shows that at a time t certain configurations were more probable 
than others, we inscribe on the blank sheet a function /(#i, 0%, ... t) in- 
dicating that the actual probability of the configuration (6 lt 2 , ...) was/ 
times the initial probability. We call / the modifying factor. The modified, 
i.e. the actual, distribution is therefore represented by a strain vector S 
which does not in general satisfy | (det Sft | = 1. The factor / is necessarily 
an algebraic function. In wave analysis it is expressed as the product of two 
wave functions, and is therefore formally a wave tensor of the second rank 
in which all terms are zero except the quarterspur. This suggests a 
generalisation of the modifying factor /. It often happens that in intro- 
ducing the modified probability we make at the same time a transformation 
of coordinates; that is to say, we compare the modified probability of 
a configuration with the initial probability, not of the same configuration, 
but of a configuration related to it by a transformation. The trans- 
formation and the modification of probability are comprised in a non- 
algebraic modifying factor/, which is the product of vector wave functions. 

Whether we are treating the initial or the modified distribution, its strain 
vector is normalised so that the total probability in the volume ii c is 1. In 
this normalisation the time dimension 16 is excluded, because the con- 
ception of distributing probability over extension in time is rather unusual. 
An atom exists continuously in time, so that the association of its pro- 
bability with particular time intervals dt does not arise in a direct way. 



108 Wave-tensor Calculus [7-7 

But consider a clock with one hand moving at a uniform rate. If the clock 
is part of the system, we shall, in specifying the configuration of the system, 
specify the position angle 16 of the hand of the clock; and the elementary 
range of configurations dot will include the element d0 16 of position angle. 
Thus from the internal point of view 16 is an ordinary angular coordinate 
in the specification of configuration, and is one of the dimensions of the 
domain over which the unit probability is distributed. It may be included 
in the general normalisation; or (owing to its commutation with all the other 
phases) it may be excluded and normalised separately, so that the pro- 
bability d0 16 /2?r of a range d0 16 is stated separately. 

It was rather surprising to find time appearing at all in phase space. We 
apparently rid ourselves of it when we retained only the displacements 
associated with space-like matrices. But it has gained entry in space-like 
disguise as the position angle of the hand of a clock. Moreover, it is a periodic 
angular coordinate, not an open infinite coordinate like external time. 

This may be made clearer by an illustration from celestial mechanics. 
The orbit of a planet is specified by six elements, one of which is the epoch 
of perihelion passage T. The element T corresponds to 16 . If there are 
several planets, we must include the element T for each of them in enumer- 
ating the possible systems which might be formed. Thus T is an essential 
coordinate of the configuration space in which we represent the possible 
combinations, although it is ordinarily conceived as having a time-like 
character. 

Since time is measured by a phase angle, instants which differ by multiples 
of a period are to be considered identical; and the whole extent of time is 
27r.fi in natural linear coordinates. That is because in the structure assigned 
to the system there is no provision for a revolution counter. A more extended 
time reckoning can only be given a meaning when we treat more complicated 
systems. It is fairly obvious that infinite time will appear automatically 
when we introduce systems with incommensurable periods. 

7*8. Physical Meaning of Strain Vectors. 

By the aid of space vectors we have defined a domain which has the primitive 
property of conceptual space-time, namely that all points of it are equiva- 
lent. It is all one whether a given object is at the point A or at the point B. 
But this is in flagrant contradiction to our experience that somehow it is 
possible to find out that the object is at A, not B. We have to combine two 
different conceptions of position absolute and relative. By absolute I mean 
4 'conceived as absolute", i.e. pictured in an abstract geometrical frame; by 
relative I mean "relative to observable landmarks". In so far as the points 
of space-time symbolise absolute positions they are equivalent to one 
another; in so far as they symbolise relative positions they can be dis- 



7-8] Strain Vectors and Phase Space 109 

criminated observationally. According to whether the point is being con- 
sidered in its absolute role or its relative role, the position vector defining it 
is a space vector or a strain vector. 

The same applies to other physical entities. The energy and momentum 
constitute a vector. Referred to an abstract geometrical frame, this is a 
space vector, but relative to another physical system it is a strain vector. 
The difference between space vectors and strain vectors might be defined 
by saying that when we treat of space vectors we are contemplating the 
" blank sheet"; when we treat of strain vectors we are beginning to write 
something on the clean page. 

It is true that when we contemplate two space vectors we are implicitly 
writing something on the clean page, namely the internal relations of a 
system consisting of two parts. But the treatment of two vectors, and the 
extraction of the internal relations of that which they represent, is slightly 
more advanced than the problem we are now handling, and is not suitable 
for our first writing lesson. We here treat the internal relations as already 
extracted and presented to us in the form of a strain vector. 

The conception of relative position or relative momentum arises when, 
instead of contemplating the particle as a solitary system in an abstract 
frame, we regard it as part of a more comprehensive physical system. 
Consider, for example, an electron in an atom. A displacement of the 
electron has two aspects. It is a translation of the electron from one point 
to another in external space-time; as such it is a change of the space 
vector defining the absolute position of the electron. But it is also a 
deformation or strain of the atom; as such it is a change of a strain 
vector. The atom is changed to a new configuration, so that there is a 
displacement in the phase space in which the configurations of the atom 
are represented. 

The tensor calculus provides a machinery for locking the changes of one 
characteristic to those of another. The changes of strain of the system are 
locked to the changes of absolute position of its parts. We should commonly 
say that the change of position of the electron is the cause of the change of 
strain of the atom. To work out this, connection in an actual system of 
several particles is too complex a problem for us at present. We proceed in 
the converse way. We consider the simplest form of tensor interlocking, and 
construct the ideal "system " to which it would apply. We cannot find any 
more elementary starting point thaji the locking of a covariant wave tensor 
to a mixed wave tensor; and the former has been called a strain vector in 
anticipation of this application. 

Accordingly when we wish to pass from the absolute to the relative 
aspect of the position of a particle, we treat the particle as an element of an 
ideal physical system whose configurations (so far as they are determined by 



110 Wave-tensor Calculus [7-8 

this particle) are specified by a simple strain vector, and therefore occupy 
a phase space of the kind we have been investigating. 

What exactly has been added to the particle to make it part of a physical 
system instead of a lone particle to de-absolutise it? Mathematically it is 
the matrix J? 45 , which transforms its space vector into its strain vector. This 
matrix defines a particular three-dimensional section of the world a plane 
of simultaneity. Physically the particle is made part of a system by asso- 
ciating with it a plane of simultaneity. 

We have seen that when a number of particles are treated as a combined 
system, each particle retains its separate space coordinates, but there is only 
one time coordinate for the whole system. This replacement of the in- 
dividual time coordinates by a common time coordinate is the essence of the 
process of combining ; it defines the change in our point of view when we 
consider the system as a whole instead of its constituent parts. A hydrogen 
atom is composed of a proton and electron; but a proton today and an elec- 
tron yesterday do not constitute a hydrogen atom. We have seen ( 7*1) that 
the planes of simultaneity which correspond to the common time coordinate 
of the system are determined by the direction of its external momentum 
vector. 

We can now see how the conversion of space vectors into strain vectors 
corresponds to our change of attitude when we consider the particle to be 
part of a system. Actually to introduce the other particles of the system 
would greatly complicate the problem; but, in anticipation of their presence, 
we introduce the planes of simultaneity which will be determined by the 
external momentum vector of the system when it appears. We construct 
these planes in the geometrical space-time which previously contained no 
indication of a particular direction of section; we inscribe the matrix J5 45 , 
henceforth to be permanently associated with the planes of simultaneity, 
on the sheet which was previously blank. Simultaneity is no longer arbitrary ; 
we cannot modify the reckoning of it to suit a particular particle of the 
system. Suppose that the particle has a momentum in the ^-direction; by 
(6-51) the space vector representing this is E l& p l . Formerly the particle 
could be "reduced to rest" by a Lorentz transformation with matrix of the 
form J5? 14 itt 14 ; but we are no longer allowed to rotate the plane of simul- 
taneity. The strain of the system (as compared with a system in which the 
particle is at rest) is measured by the inhibited rotation E^iu^ . Apart from 
a numerical factor this is the strain vector E l5 p l .iE^ 5 = E 1 ^ip l associated 
with the space vector E-^p^ 

This example calls attention to another feature of the connection between 
internal and external space. The external velocity or momentum vectors 
correspond to internal displacement vectors, and vice versa. At present we 
can only recognise this in a preliminary way; but another example may be 



7-8] Strain Vectors and Phase Space 111 

of interest. If we apply this principle to the internal energy attached to the 
matrix E 19 , the corresponding space vector attached to the matrix 7S 45 
should be interpretable as an external time displacement. As in the previous 
example it will be an inhibited or virtual time displacement. If we change 
the scale of the system, we change the light-times between its various parts, 
and therefore change the time at which a particle becomes causally effective 
at the centroid where the external wave vector is supposed to be located. 
But in our internal wave functions the particles are assigned simultaneous 
times, not the times at which they are causally effective. Thus the distance 
of a particle from the centroid can be looked upon as an inhibited time displace- 
ment. As an isolated particle we should have contemplated it at an ante- 
dated instant to allow for the lag of causal efficiency; as part of a system we 
can only contemplate it at the same instant as the rest of the system. If it 
receives a radial displacement 8r, a further lag 8rjc is introduced which (if the 
particle were not considered to be part of the system) would be compensated 
by giving it a time displacement 8r/c. Since the planes of simultaneity in 
the system are fixed, we cannot give an individual time displacement to the 
particle; and the inhibited time displacement appears as, and is the measure 
of, a strain of the system. We have seen that this strain will be associated 
with the matrix jE7 16 and therefore be an internal energy. Thus a system will 
in general have an internal energy depending on its linear scale. This is one 
aspect of the origin of electrostatic energy .f 

Our present point of view is that, instead of starting with an elementary 
particle defined to be such that its properties can be represented by a com- 
plete space vector, it is rather less abstract to start with an elementary 
system defined to be such that its condition or "configuration" can be 
represented by a simple strain vector. Then by introducing the associated 
space vector we detach the active principle of the system from the passive 
principle represented by planes of simultaneity, and so obtain a still more 
abstract entity, namely an elementary particle in free space. It may seem 
far-fetched to describe a particle coupled with a plane of simultaneity as a 
"physical system". But that is as much of a physical system as we can 
represent by a simple wave tensor. And inasmuch as current quantum 
theory has made shift to treat a wide range of problems of observational 
importance with simple wave tensors, it is an important stage in the advance 
of the theory towards actuality. 

If a system A were completely isolated it would be unobservable. There- 
fore it enters into observation as part of a more extended system B. As such 
it will be represented in the internal space of B. The external momentum 

t Conversely, if we derive the electrostatic energy as in Chapter xv, the foregoing 
investigation shows how to connect with it the idea of antedating the particles by the 
light-time. 



112 Wave-tensor Calculus [7-9 

vector of B determines planes of simultaneity for the internal constituents 
of B\ and the external momentum vector of A, which was a space vector so 
long as A was considered in isolation, is replaced by the associated strain 
vector when A is considered as part of B. But B in its turn would be un- 
observable if it were not part of a larger system C\ its external momentum 
vector must therefore be replaced by a strain vector, representing B as a 
constituent of the internal structure of C. And so on. 

At each stage there is an external space vector of the system contemplated, 
which is converted into a strain vector when we consider a larger system 
until the system has been extended to comprise the whole universe, when 
the external vector can be dropped, all phenomena being comprehended in 
the internal structure. 

From this point of view space vectors make only a temporary appearance 
when we halt for breath in the course of the analysis. As soon as we are 
ready to proceed further, we replace them by their associated strain vectors. 
There are, however, two reasons why space vectors remain an important 
conception in practical investigation. Firstly, it is impracticable to treat 
the exact equations of the internal statew of highly complex systems; 
therefore when we have to deal with'a number of systems with weak inter- 
action, we leave them uncombined and treat their mutual influence (which 
brings them within reach of observation) by approximate perturbation 
methods. Secondly, in dealing with a large number of similar systems, we 
combine them statistically, not individually. The external space vectors are 
not transformed individually into internal strain vectors of the complex 
system; but are first replaced by a probability or average distribution. The 
internal state of the complex system is described by a very much simplified 
set of strain vectors embodying coefficients of the distribution function. 
Thus the internal state of a gas is described by pressure, energy density, 
virial, vorticity, etc., representing certain averaged characteristics of the 
external space vectors of the molecules. Normally the direct procedure of 
replacing space vectors by their associated strain vectors is not extended 
to systems greater than a molecule.f 

7-9. Singular Phase Space. 

The phase space which we have been considering is generated by the trans- 
formations of a non-singular strain vector. If we use instead a singular 
strain vector S a , the resulting phase space has fewer dimensions, since 
S a dQ 8 s=Q when d 8 is a pseudo-reciprocal of S a . Displacement in such a 
direction involves no change of the strain vector and therefore no change of 

t The averaging could perhaps equally well be performed on the associated strain 
vectors. But since space vectors are more familiar we generally adhere to them as long as 
we can. 



7-9] Strain Vectors and Phase Space 113 

configuration; the direction accordingly is not a dimension of the phase 
space of S a . 

To show the relation between singular and non-singular phase space, we 
consider a non-singular strain vector which at some point of phase space has 
a purely algebraic value S . By (6-64) this is analysed into spectral coiu- 

P nentS = S . (7-91) 



The components are pure and therefore singular. Consider the component 
S a . Since the products S a 8 b , etc. are zero, the singular directions, which 
satisfy S a d& 8 = 0, are ^ = 6/g& + ^ + dSd (7 . 92) 

Since d 8 cannot contain time-like matrices, the ratio b:c:d must be chosen 
so as to eliminate the time-like matrices J5? 23 , J57 45 . There is therefore only 
one singular direction d&.=c(8 + 8 d ). (7-93) 

If, as in (6-64), S f , = - iiS (#1 + # 23 + #45 + # 16 ), (7-941) 

the singular direction is d 8 =(-E l + # 16 ) d<f>. (7-942) 

For S c and S d the singular direction is d& 8 = (E l -f E 19 ) d<f). 

Non-singular phase space of ten dimensions can thus be regarded as a 
superposition of two singular phase spaces of opposite spin, each of nine 
dimensions. The axis of spin, here represented by E t , can be in any direction 
in three-dimensional space. When the singular phase space of 8 a is delineated 
in ordinary phase space, the added dimension (direction of </>) gives a line of 
indistinguishable configurations which are really one configuration. The 
equation of the singular line (7-942) may be written 

d<f> = cZ0j_ = dd^Q . 

Or, since dt = Rd6 IB , dOJdt = - I/ R. 

The singular line therefore represents an entity spinning with uniform 
velocity. We do not distinguish the different orientations corresponding to 
the sequence of points along the line, but count the state of spin as one con- 
figuration a constant state of strain of the system. The strain produced by 
the rotation is to be compared with the gyrostatic torque of a fly-wheel, not 
with the torque of a wound-up spring. A change of the plane of spin would 
be a change of configuration. 

When we separate non-singular phase space into two singular phase 
spaces, we require an additional variable to specify how the probability is 
divided between the two phase spaces. This extra variable compensates for 
the loss of a dimension in passing to singular phase space. It is to be remem- 
bered that S a + S c does not represent two particles of opposite spin; two 
particles would require a double wave tensor. It represents a particle which 
has equal probability of having either spin. 



114 Wave-tensor Calculus [7-9 

It is perhaps rather surprising that there is no corresponding separation 
of phase space for positive and negative charges. The physical reason appears 
to be that the conception of phase space is bound up with the conception of 
statistical equilibrium, and there can be no equilibrium if the charges (or 
the probabilities of positive and negative charges) do not balance. Equi- 
valently it is attributable to our result that electrically saturated space 
satisfies different reality conditions; and the phase space (if any) associated 
with it would require to be reinvestigated from the beginning. Although 
there is considerable analogy between opposite sign of charge and opposite 
spin, there is the fundamental distinction that opposite charges, unlike 
opposite spins, cannot be transformed into one another by relativity 
rotations. 



CHAPTER VIII 
THE DIFFERENTIAL WAVE EQUATION 

8* 1 . Conservation of Probability. 

In the terminology introduced by Diracf a "state" of a system consists of 
a particular distribution of probability over the various possible configura- 
tions. The state is supposed to extend over all time. The probability dis- 
tribution may vary with , but always so that the integrated probability of 
all the configurations for a fixed value of t is unity. In other words pro- 
bability is conserved. 

We may treat the probability as a fluid occupying the configuration space, 
the probability of a given range of configurations being represented by the 
mass of fluid in a corresponding volume. The change of distribution of 
probability is then represented by a motion of the fluid. 

The method of wave mechanics is to analyse the whole probability of the 
system, which must be unity, into the probabilities p a , p b , p c , . . . of a set of 
elementary states a, b, c, .... Then if ^(a^, t) is the probability of the con- 
figuration Xp at time t in the state a, the whole probability of the configura- 
tion X at time t is 



A. perturbation of the system means a variation ofp a , p b9 p c , ..., subject to 
their sum remaining unity. 

The object of this device is to separate the mathematics of interaction 
from the mathematics of structure; for the influence of extraneous bodies is 
described by changes of the factors p, and the functions q which describe 
the structures of the various states remain unaffected. 

Since the elementary states are introduced for analytical purposes, we 
may impose on them such limitations as appear advantageous. It would be 
possible to contemplate discontinuous flow of probability, whereby pro- 
bability is created at one point and disappears at another point in the state, 
subject to the total probability remaining constant; but in current quantum 
theory the states are assumed to be continuous, that is to say the fluid moves 
subject to the equation of continuity. 

We introduce a vector j^ whose component in the time direction gives the 
density of the probability fluid and whose space components give the 
density of its flux. The conservation of probability is then secured by the 
equation of continuity of the fluid 

div^-0. (8-12) 

t Quantum Mechanics, 1st ed. 



116 Wave-tensor Calculus [8-1 

This assumes that the configurations are represented in a space with Eucli- 
dean metric. More generally the density of probability and of its flux are 
represented by a vector density j^, and the equation of continuity is 

divj^O, (8-13) 

the covariant divergence of a vector density being the same as its ordinary 
divergence. 

For the general development of the theory it is necessary to use the phase 
space and strain vectors treated in Chapter vn; but in order to connect our 
treatment with Dirac's theory, we first consider a simple particle whose 
"configurations " are determined by its coordinates x, y, z in ordinary space. 
The probability distribution in state a is then a function q a (x, y, z, t). 

Since we admit perturbations, we have to recognise an independent 
variable s extraneous to the state as an argument for the perturbations, so 
that (8-11) becomes I^p. <).*. (a, y.M). <*14) 

The distinction between s and t is often ignored; but it is necessary to attend 
to it when we consider Lorentz transformations. We may apply a Lorentz 
transformation to x, y, z, t, and thus obtain a new but equivalent description 
of the probability distribution in the state; but the function p a (s) is a factor 
applying to the whole domain of x, y, z, and the time s which appears in it is 
not associated with any particular values of x, y, z; there is therefore no 
possibility of applying a Lorentz transformation to s, and it is invariant 
for the internal transformations of the state. 

The perturbations are imposed on the system from without, and occur at 
times fixed according to an extraneous time reckoning. If we like, we may 
choose the time t within the state so as to conform to this reckoning; but we 
then lose the possibility of applying Lorentz transformations to the state. 
It is undesirable to do this at any rate at the present stage since the 
study of the Lorentz transformations, initiated by Dirac, has contributed 
greatly to our understanding of the theory. To preserve this advance and 
exhibit it in its proper relation to the more general developments, we have 
to distinguish a relative time t in the state and an invariant time s which 
serves as a link with other systems.! 

If we wish to make a relativity transformation which includes both the 
perturbed and the perturbing system, we must treat them as one combined 
system and analyse the probability distribution into elementary states of 
the combined system. But then, for this combined system, there will be an 
invariant time 8 forming the argument of the perturbations of the combined 
system by systems extraneous to it. However many systems we combine 
there is always an invariant time left over for the purpose of representing 

t In the nomenclature introduced later in this chapter, t is a geometrical coordinate and 
a a dynamical coordinate. 



8-1] The Differential Wave Equation 117 

the perturbation effects of systems not yet included. It would be idle to 
consider a system without making provision for perturbation from outside; 
for in that case we could never acquire knowledge of it unless indeed it 
were so extensive as to include the brain of an observer. 

We note in particular that the time t is reversible, but the time s is irre- 
versible. The axes in the state can be rotated so that ->- 1, and there is no 
absolute distinction between waves travelling forward and waves travelling 
backward in t. The latter are oddly said to have negative energy or mass 
according to the technical definitions in quantum theory. But the inter- 
action with other systems depends on the invariant time s\ thus the rotation 
of t and the substitution of "negative mass" for positive mass does not 
signify an observable change it would not be a relativity transformation 
if it did. Similar considerations apply to more complex systems, until we 
reach a system which includes a human brain and is observed from within 
instead of from without. Extraneous time 8 is then no longer needed. But 
the inclusion of the observer in the system automatically prevents the 
relativity transformation from being pushed so far as to turn t into - 1\ at 
least it is unusual to include among the admissible systems of description 
(1-1) those of an observer whose consciousness runs backward in the 
adopted time reckoning, and who endeavours to predict the past from his 
memories of the future. 

By (8-14) the probability of a configuration x^ is a function of the two 
times s, t. The same duality of time occurs, for example, in perturbation 
theory in celestial mechanics; at each moment s there is an osculating orbit 
of a planet which professes to give the position of the planet for all times t 
between - oo and + oo. These positions are not actually realised. To obtain 
the realised positions we have to associate corresponding values of s and t. 
This means that in the present problem we have to lay down in the state a 
series of locif(x,y,z,t)=s which give a "representation" of s in the state. 
In particular cases it could be arranged that f(x,y,z 9 t) = t = 8', but, as 
already stated, this definite choice of t requires that we forgo the applica- 
tion of Lorentz transformations. 

The coordinate s, which is primarily.an extraneous time-variable but has 
also a representation in the state, is a connecting link between the system 
which is being described and the rest of the universe. This ideal type of 
connection cannot be exactly realised in practice. The approximation lies 
in the assumption that, whatever the source of the perturbation, its argu- 
ment s is represented by the same series of loci/ (x, y, z, t) in the state. Prac- 
tically this means that the velocity of propagation of all perturbations is 
assumed to be infinite. If we require greater accuracy, the only course is to 
amalgamate the perturbed and perturbing system and investigate the 
states of the combined system. 



118 Wave-tensor Calculus [8-1 

To insist on this greater accuracy would be a counsel of despair. It would 
be to tell the physicist that, since the whole universe is interrelated, it is no 
use his attempting to study anything less than the whole universe. It is 
obvious that it is profitable to study portions of the universe as isolated 
systems. The method of states provides for this, and at the same time makes 
provision for re-attaching these systems to the rest of the universe in an 
approximate way, so that they may not be entirely cut off from observation. 

It would be possible to choose the set of elementary states a, b, c, ... in 
many different ways. In practice therefore certain limitations are imposed. 
In particular the states are so chosen that the wave tensor SJ2^ j^ corre- 
sponding to the probability vector is factorisable. The state is then said to 
be pure. Further, the whole set of states must be complete without being 
redundant so that, when the probability distribution of the configurations 
at time t is given, the coefficients p n , p b , ... are uniquely determined by it. 
The conditions of purity, completeness and non-redundancy require that 
the states shall correspond to a spectral set of operators. The spectral 
set is, of course, much more general than that which we have introduced for 
resolving an isolated wave tensor ( 5-7). We shall not enter into the detailed 
treatment of these conditions, since our theory here coalesces with current 
quantum theory. 

8* 2 . The Divergence Condition . 

For a system describable by simple wave tensors the probability vector j^ 
representing a pure state will be of the form 



We introduce two divergence operators 

16 

V-S^a/eteg, V 
where 8/80;^ signifies 3/9^ written after its operand. Then sincej" = 



16 16 

S^a/eteg, V* = 2(8/8^)^, (8-21) 



so that, summing for /A= 1, 2, ..., 16, 

divj = S (dj^dxj = - Jx* ( v * + v ) 0- (8-22) 

It is understood that the same frame E^ is used throughout the whole 
domain of x^ in which and x* extend. 

The result of performing the operation V on $ is another four-valued 
quantity o>. We can always find a matrix M such that co = M ift. Generally 
M will be a function of the coordinates. We therefore set 

Vt = Mif>, x*V* = x*M*, (8-23) 

so that div j = - ** (M* + M) 0. (8-24) 



8-2] The Differential Wave Equation 119 

We ensure the vanishing of divj by setting Jf*= If; so that (8-23) 
gives the differential wave equations 

(V-Jf)^ = 0, x *(-V*-Jlf) = 0. (8 . 25 ) 

These wave equations will be invariant for all wave-tensor transforma- 
tions, if V, V* and M are mixed wave tensors. This requires that 3/3a^ shall 
be a complete space vector; that is to say, when a relativity transformation 
is applied, it is transformed into a new array of operators 3/3^' which are 
linear functions of d/dx^ , just as if it were a numerical vector. The condition 
that 3/Sa^ is a complete space vector is equivalent to the condition that x^ 
is a complete space vector. 

If we limit ourselves to solutions of (8-25) which are functions of four 
rectangular coordinates (x l9 x 2 ,x 3 , # 4 ), the equations reduce to 



s <8 ' 261 > 



These equations are invariant only for the six relativity rotations in four 
dimensions; because the other relativity transformations (applicable to 
(8*25)) would reintroduce the terms that have been dropped. 

Dirac's wave equations for an electron of proper mass m in an electro- 
magnetic field, whichgivesit an energy and momentum K^ , can be written as 



(8-262) 
Comparing (8-261) and (8-262), we obtain the identification 



(8-271) 
Writing as usual M = S m^ E^ , we have 

m le = m (wij , w 2 , w 3 , w 4 ) = - i fa , K Z , /c 3 , * 4 ). (8-272) 

We have made this comparison with Dirac's equation in order to ascertain 
the current nomenclature for the components of our space vector M. The 
consequences of his equation have been worked out and compared with 
experiment, so that we know how the quantities m, K^ contained in it 
manifest themselves observationally. Equation (8-272) transfers this 
knowledge and the consequent nomenclature to our own equations (8-261). 
It would be foreign to our plan to intermingle the current semi-empirical 
theory with the purely deductive theory that we are developing, and our 
reference to the current equations is for identification purposes only. In 
current theory m is supposed to be a constant independent of the coordinates. 



120 Wave-tensor Calculus [8-2 

We have as yet shown no reason why m should be constant; that is a con- 
sequence of the dynamical equations, and will be proved in 9-2; but it will 
be assumed here in anticipation. 

If ^-iJL + ^i^.^, (8-28) 

(8-262) reduces to 



which formally agrees with the simple wave equation as given in (5-13), 
(5-14). But this is only a real agreement if (8-28) gives an algebraic value of 
Ppl for in (5-14) the p^ are necessarily algebraic coefficients. 

In order that the p^ defined by (8-28) may be algebraic, and x* must be 
eigensymbols of d/dx^ . This requires that shall be a function of x^ of the 

form Jf(x x x. x ) = /(# x x x ) Ji (8-285) 

where / is an algebraic function and a constant wave vector. We shall 
call wave functions of the form (8-285) algebraic wave functions. 

It would be undesirable to exclude non-algebraic wave functions. We 
shall find, for example, that the wave functions giving the steady states of a 
hydrogen atom are non-algebraic. Thus for unrestricted wave functions the 
momenta p^ are non-algebraic ; and the identification of p with the algebraic 
coefficients^ in 5-3 does not apply. 

Thus in general we have two independent equations 

= 0, (8-291) 

= 0. (8-292) 

The first is an identity, except that it is implied that the axes are so chosen 
that J 5 = 0. The second expresses the conservation of probability. For 
algebraic wave functions the two equations are the same; so that the 
momentum vector is the same as the stream vector except for a numerical 
factor. For non-algebraic wave functions the two equations are distinct; 
and no comparison between the stream vector and momentum vector is 
possible, since the components of the latter are non-algebraic quantities. 
We can rewrite (8-292), setting 

4 16 



where the p ' are algebraic coefficients; then the momentum vector can be 
regarded as a complete space vector p^', which has been formally reduced to 
four dimensions by the device of admitting symbolic components. 

If there is no electromagnetic field, (8-262) has the elementary solutions 

^ (8-293) 



8-3] The Differential Wave Equation 121 

where the p^ are algebraic, and ^ , Xo* are solutions of the elementary wave 
equations (5-13). 

In this book we shall use non-algebraic wave functions in special in- 
vestigations, but in pursuing the more fundamental problems we shall 
generally limit ourselves to algebraic wave functions. That is because we are 
especially concerned with the borderland of relativity and quantum theory, 
and, as explained in the introduction, their meeting-point is to be found in 
the most uniform conditions. 

According to (8-28) there are two different formulae for y^ . At present the 
position (final or initial) of the wave vector is a sufficient indication which 
formula applies. But it is hampering to have a definition of p which pre- 
vents us from changing the position of the wave vector; and we shall later 
describe the difference of the two operators in a more general way ( 8-6). 

8- 3 . Govariant Differentiation . 

We do not propose to generalise our formulae to apply to all kinds of curvi- 
linear coordinates; but there is one kind of curvilinear coordinate which it 
will be necessary to use, viz. an angular coordinate. Consider the trans- 
formation from rectangular coordinates (x l9 x 2 ) to polar coordinates (r, 12 ) 
in a plane; the question arises how this change is supposed to affect a general 
space vector jL and its wave- vector factors ^r, x*. 

Our theory of has been based on an orthogonal frame of reference, and 
it would be a grave complication to depart from an orthogonal frame. 
Therefore we do not consider the polar components of a vector in the sense 
of general relativity, but rather in the sense of elementary mechanics in 
which the "polar components" of a force are its rectangular components in 
the radial and transverse directions.! That is to say, our transformation 
will correspond to (dx^ , dx 2 ) -> (dr, rd0 12 ), not to (dx^ , dx 2 ) -> (dr, d0 12 ). Thus 
we retain an orthogonal frame, but the frame rotates as 12 changes; and a 
local vector 0x* * s resolved orthogonally in continually changing directions. 

We shall now find the covariant derivative of iff with respect to 12 . By 
(4-15) the transformation $' = C* AI ^" gives J/ = ^ cos 12 - 1 2 sin 12 > which 
is the change of t due to the axes having been rotated through an angle 
12 . In the present case, when the angular coordinate changes from to 
12 , the axes are rotated in the same direction so that the corresponding 
transformation of is y = e -+* u e^ m (8-31) 

This assumes that there has been no "real" change of 0; that is to say, if we 
had kept to rectangular coordinates, ^ would have had the same value at 

t There is a distinction between "a transformation of coordinates from (a;, y) to (r, 0)" 
and "a transformation from rectangular coordinates (x, y) to polar coordinates (r, 0)". 
The first implies that all coordinates are to be treated alike as in general relativity; the 
second implies that they are to receive the distinctive treatment usually accorded to rect- 
angular and polar coordinates. (Of. Mathematical Theory of Relativity, 16.) 



122 Wave-tensor Calculus [8-3 

(r, 12 ) as at (r, 0). We call displacement without real change parallel dis- 
placement. Differentiating (8-31), the change due to parallel displacement is 
3^7^12 = ~"i^i2^'- For a general displacement we obtain the covariant 
derivative by subtracting from the apparent change d*fi'/dO l2 the change 
arising by parallel displacement. Thus the covariant derivative operator is 

>. (8-321) 

(r\ \ / 7\ \ 

- - - ) = - 1 1 - + p? y | (8-322) 

vUpv/c \vOpv / 

is called the angular momentum conjugate to the coordinate 6^ v . The term 
\iEpy is called the spin momentum. It was originally discovered as a 
correction to the angular momentum, which had previously been assumed 
to be id/ddpy . We see that the spin momentum is merely the difference between 
the covariant derivative and the ordinary derivative. It is a nominal addition 
to the angular momentum due to our non-relativistic outlook. 

By (3-38) the expectation value of E^ is ij^/jw, so that the expectation 
value of the spin momentum is ^j^/j^. The term "spin momentum" is 
primarily limited to the components j M , j 31 , j 12 ; but analytically all 
components are on the same footing, and we have a complete space 
vector //2j 16 giving the part of the momentum (expectation value) which 
arises from the difference between covariant and ordinary derivatives. 
From this aspect the stream vector J is also a momentum vector. The 
identification in 6-6 of its E^ component with mechanical spin is thus 
confirmed and elucidated. 

For an initial contravariant wave vector #*, the corresponding operators 
are 



(8 ' 332) 

Hence the covariant derivative of a space vector J = ^^* is 



, + * (E^ J - JEy,). (8-34) 
The angular momenta generally referred to in quantum mechanics are 
conjugate not to the angles 6^, but to angular parameters - ^ v introduced 
in the following way. Consider a distribution of ^ constituting an elementary 
state of a system. If the boundary conditions, extraneous electromagnetic 
fields, etc. are symmetrical in the plane of 0^ , we obtain another elementary 
state by rotating the whole distribution of through an angle a F in that 
plane. Thus we obtain a series of distributions iff(x l9 x 29 x 9 ,Xt, V) *i> *i 



8-3] The Differential Wave Equation 123 

x a , # 4 being coordinates, and a^,, a parameter distinguishing one distribution 
from another. In place of x ly x z , x 3 , # 4 we can use polar coordinates, in- 
cluding the angle B^ v . We have 

since the first term in the exponential gives the effect on $ of rotating the 
axes backward through a^,,, which is equivalent to rotating the whole dis- 
tribution forward through oc^, and the second term is introduced because 
the point 0^ in the new distribution corresponds to the point 6^ v - oc^ in 
the old distribution. 

Differentiating (8-35) with respect to a^, we have 

(8-361) 



and the angular momentum conjugate to oc^ is 



For x* the corresponding operator is 

(8-363) 



We shall call a^,, a dynamical coordinate, and distinguish M^ and M IW as 
geometrical and dynamical angular momenta, respectively. The importance 
of M v is that (in the symmetrical conditions already postulated) it com- 
mutes with the hamiltonian; this, as we shall see later, makes it a constant 
of the motion of the system, which is then said to possess an integral of 
angular momentum. But when the conditions are not symmetrical, and no 
integral of angular momentum exists, there is no reason to suppose that 
(8-362) represents angular momentum of any kind. It is therefore rather 
misleading to say that -*(3/3fl^-i^ f f4 J is the angular momentum of a 
system; it is a form to which the dynamical momentum reduces in parti- 
cular cases when it happens to be constant. On the other hand, M^ has 
the same interpretation for symmetrical and for unsymmetrical systems. 
Whilst it has apparently little connection with momentum as conceived 
in classical mechanics, it is a natural generalisation of the quantum theory 
definition. 

To reach the dynamical outlook we must promote <x 12 , originally intro- 
duced as a parameter, to be a coordinate; so that ^ is a function of five 
coordinates <x 12 , x l9 x 2 , x z , # 4 , or in polar coordinates a 12 , 12 , <, r, t. (The 
configurations occupy four dimensions 12 , <, r, t, as before.) This means 
that a series of elementary four-dimensional states is run together to form 
a single five-dimensional state; and we adopt a new dissection into states 



124 Wave-tensor Calculus [8-3 

in which the five-dimensional state is regarded as elementary .f In combining 
the four-dimensional states we attribute to them a uniform probability 
distribution in a 12 . We can now assign only one probability factor p to the 
whole five-dimensional state, whereas formerly we could assign different 
probability factors p a to each orientation a of the four-dimensional state. 
This sacrifice corresponds to the fact that, the conditions being perfectly 
symmetrical, it is impossible to distinguish the orientation observationally, 
and therefore we never have occasion to consider a modification of the 
initially assumed uniform probability distribution in orientation ("a priori 
probability") through additional information furnished by observation. 

The usual method of obtaining (8-362) is to consider a form of hamiltonian, 
which by its symmetry ensures that the dynamical angular momentum is 
constant, and then to identify an analytical expression which turns out to 
be constant as the angular momentum. This gives no indication of the ex- 
pression for a non-constant angular momentum. For example, in an asym- 
metric electromagnetic field, systems, besides being deformed by the field, 
will tend to orient themselves in a certain way. Even if we ignore the 
deformation and assume that exactly similar four-dimensional states can 
exist in different orientations a, it will be necessary to insert in (8-35) a 
probability factor p^ representing the unequal probability of distribution 
of the different orientations; so that there will be an additional term in 
30/8a. It is clear therefore that (8-362) is not the correct expression for the 
angular momentum in unsymmetrical conditions. 

In setting divj = S Sj^/dx^ ( 8-2), we assumed that the x^ are rectangular 
coordinates, the volume element being taken to be dV = dx 1 dx 2 dx s ... so 
that the probability or probability flux belonging to an element isj^dV/dx^ . 
If angular coordinates are used, two courses are open. The simplest is to 
introduce the vector density j^j^V -g; the equation of conservation is 
then (dj^/9^) = 0. Otherwise we must substitute covariant derivatives 
in place of ordinary derivatives in V. 

It is instructive to check the agreement of the two methods. If E f is the 
matrix corresponding to the radial direction, and E e the matrix for a rota- 
tion in a plane containing the radius (therefore anticommuting with E r ), the 
matrix for the corresponding transverse direction is Eg E r . The corresponding 
term in V is 



-** + - (8 ' 37 > 

t The running together of the configurations along a singular line in 7-9 to form a single 
"state of spin" of an elementary particle may be regarded as an elementary example of this 
procedure. 



8-4] The Differential Wave Equation 125 

Thus for each rectangular coordinate replaced by an angular variable, 
E r /2r is added to V. If there are n angular variables, the result is to change 

the radial term E r dldr to 

""" (8-38) 



If we set 0=r * n ^ and take ^ as a new wave function, the extra term 
\n\r is eliminated. We find that x* behaves in the same way; so that, setting 
X* = r~* w co*, the extra term is eliminated in the equation for o>*. (In 
treating #* we must notice that Eg E r is replaced by E r Eg , and this cancels the 
change of sign of the spin correction.) Accordingly we have a new com- 
bination 3 = <!><*>* = r n J which satisfies the simple divergence equation with- 
out the added terms. This will agree with our former result if 3 is the vector 
density JV^gi hence V g should be equal to r n . This is correct, because 
each substitution dx^rdO^ contributes a factor r to V g. 

8-4. General Dynamical Equations. 

Since our symbolic calculus has been extended by the introduction of 
differential operators djdx^ , which I will call Z)-symbols, it is necessary to 
refer again to the formal definitions in 2-1. 

A symbol which commutes with every symbol in the calculus will be 
called an algebraic number as heretofore. A symbol which commutes with 
all symbols other than D-symbols will be called an algebraic function.^ 

Since D ($x) ^ (D$) x, the i>-symbols do not obey the associative law of 
multiplication. IfD x = d/dx, DJx = 9 (fx)ft x - Hence 



whatever x ma y be. We have therefore in all cases 

Sfldx=DJ-fD c . (8-41) 

The introduction of Z)-symbols leads us to contemplate a wider variety 
of tensor transformations, i.e. a wider variety of systems of description of 
a physical system. We have been using transformations of the form g = e w , 
where is a matrix or more generally an JS-number. It is natural now to 
admit still more general transformations in which may be any combination 
of symbols, including D-symbols. In particular we consider the trans- 
formation q== e iW8 , (8-421) 

where W is any symbolic expression and s is an algebraic parameter. 

Let x*> $ be initial and final eigensymbols of W, the eigenvalue m being 
the same for both, so that 

X*(JF-m) = 0, (TF-w)0 = 0. (8-422) 

t The term ''algebraic function" does not include "algebraic wave functions", defined in 
(8-285) as the product of an algebraic function and a constant wave vector. 



126 Wave-tensor Calculus [8-4 

Transforming these as contravariant and covariant wave vectors respec- 

tively, we have *> _. * e -**r ^ * e -ims 

x x * ' (8-423) 



so that, if J = 0**, . J' = J. 

If T is any other mixed tensor of the same class as J, 

T' = e* w *Te~ iw *. (8-424) 

Hence for an infinitesimal change ds 



so that dTlds = i(WT-TW). (8-43) 

To apply this, we regard s as a coordinate, and consider a physical system 
S described by mixed wave tensors T or the equivalent space vectors. When 
(8-43) is satisfied, displacement along s is parallel displacement of the system; 
that is to say, the tensors describing the system are unchanged except that 
they undergo a common transformation which we interpret as a trans- 
formation of reference frame. 

It is important to understand the significance of this association of ci 
tensor transformation with every displacement. If the system S is the only 
system contemplated, a change of the system of description has no useful 
purpose; but then it is idle to talk of displacing the system, since there is 
nothing to which the displacement can be referred. It is therefore pre- 
supposed that there exists besides S a reference system S' to enable the 
displacement to be recognised; and the displacement constitutes an in- 
trinsic alteration or strain of the combined system 8, 8'. The transformation 
of the system of description, applied to S but not to /S', expresses the fact 
that although S is intrinsically unchanged, its relation to 8' has been 
altered. 

In general relativity we are familiar with this change of the system of 
description which necessarily accompanies every observable displacement. 
It is expressed by the fact that an extended system of rectangular co- 
ordinates is impossible, or equivalently that the curvature of physical space 
cannot vanish. The effect of curvature is that displacements in different 
directions do not commute. In symbolic calculus we are indifferent to 
geometrical pictures, and express the same thing more directly by associating 
non-commuting operators with the displacements. The change of description 
is directly associated with non-commutation of operators, as may be seen 
from (8-43); if the operator W associated with the displacement s commutes 
with all the tensors T of the physical system no change of description 
occurs. 

The displacements whose associated operators are ^/-numbers can be 
represented in Riemannian space. It is, I think, improbable that the 



8-4] The Differential Wave Equation 127 

displacements associated with the more general operators now admitted 
are representable in Riemannian space. If we wish to represent them 
graphically we must adopt whatever form of geometry is necessary to 
provide representation of their non-commutative relations. In practice, 
however, we are not much concerned with the totality of transformations 
of the type (8-421); we have only to pick out a few special forms which yield 
comparatively simple systems.! 

The coordinate s is a dynamical coordinate of the same type as the co- 
ordinate aip V introduced in 8*3. It will be remembered that we there con- 
sidered a probability distribution in space-time and changed its orientation 
through an angle a.^ . Since the distribution was intrinsically unaltered, this 
was a parallel displacement. The space vectors describing the probability 
distribution at all points of the system underwent the same transformation 
(8-35). Since for any wave vectors ^r, #*, the infinitesimal transformation is 



we have W = - i d/fo = i 8/85, (8-44) 

so that W is the momentum conjugate to s. It will be seen that M^ and oc^ 
form a particular case of W and s. 

The distinction between geometrical and dynamical coordinates is rather 
obscured by the fact that in the most familiar system of coordinates, viz. 
rectangular coordinates, the conjugate momenta are the same. The distinc- 
tion is necessary in angular and other curvilinear coordinates because the 
conjugate momenta differ, and indeed are scarcely comparable in concep- 
tion. Further, by generalising our operators, we have introduced dynamical 
coordinates which may not be representable in the same space as the 
geometrical coordinates and may therefore have no counterpart in Rieman- 
nian geometry. The essential difference is that the geometrical coordinates 
express the internal relations of a system or probability distribution, and 
the dynamical coordinates express its relations to external objects. Geo- 
metrical coordinates are internal; dynamical coordinates are external. In 
elementary theory the only changes of external relations contemplated are 
those corresponding to change of position or orientation; but by means of 
general dynamical coordinates we can introduce external variables which 
more closely correspond to the internal strain vectors of the extended 
system which comprises the external reference objects. 

Equation (8-43) is a point of junction of the present theory with 
Dirac's theory. He arrived at it by seeking what he regarded as the 
most natural adaptation of the classical equations of motion to quantum 
conditions. 

t E.g., the forms W, U 19 U s , U a in 9-2. 



128 Wave-tensor Calculus [8-5 

8*5. Extension to Four Dynamical Coordinates. 

Let W, U L , U% 9 U 3 be independent symbolic expressions which mutually 
commute. We may expect to be able to find a common eigensymbol for 
them. It was shown rigorously in 3-7 (e) that a common eigensymbol can 
be found if the commuting symbols are matrices, and the same proof applies 
if they are symbols which satisfy algebraic equations. It is not clear that it 
applies to operators such as d/dx which have an infinite number of eigenvalues . 
But no inconsistency can arise through postulating a common eigensymbol 
for mutually commuting symbols, and the only limitation on the invention 
of symbols is that we must not ascribe to them properties which are not self- 
consistent. It would seem therefore that, if we cannot find a common 
eigensymbol for W, U l9 U 2 , C/ 3 , we are at liberty to invent one just as we 
have invented a square root of 1 in algebra. 

For us it is sufficient that there exist important applications in which 
initial and final eigensymbols $, x* of the four operators can be found. An 
example will be given in 9-2. Let the eigenvalues be w, p, l9 //, 2 , /* 3 . Then 
the transformation (8-423) can be extended to 

^' __ gi^+tfiSi-t-l^+t^s) ^ _ e lXw*4Vi l * 1 +fl2*2+/*3*3> iff. (8-51) 

As before the product e/ = ^x* is unaltered by the transformation. For 
other mixed tensors T we have in place of (8-43) 

dTjds = i(WT-TW), dTfa^i^T-TUJ, etc. (8-52) 

and W, U l9 U 29 Uz are the momenta conjugate to the dynamical coordinates 
s 9 s l9 s 29 s 3 . By (8-52) they are constant over the whole domain of (s 9 s l9 s 29 s 3 ) . 

We shall call the four-dimensional domain of the coordinates s, s l9 s 2 > 5 a 
an $-space. It is a very simple kind of space perhaps simpler than 
Euclidean space but it is unfamiliar since the axes in it are antiperpen- 
dicular. All displacements in it commute, so that it is pictured as flat; but 
in other respects it is not comparable with the space of ordinary conception. 

An elementary example of an S-space is obtained by taking 

W, U 19 C/ 2 , */ 3 = (E IQ9 E^ Ep 9 E y )m, 
where J5? a , Ep 9 E y form an anti-triad. We have then for all wave vectors 

t(8, *, *0, 5 y ) = e< ie*+^^^^+ A Vy)^^ , (8-53) 

which reduces to $(s 9 * a , s f , 5 y ) = e fw ^ ^ ^ v>0 , (8-54) 

when is a common eigensymbol of E (X9 Ep 9 E y . 

Alternatively we can describe the same domain by spectral coordinates 
s a9 s b9 s C9 s d conjugate to momenta J tt , J b9 J C9 J d defined as in (5-71). 

In the most elementary problems we cannot have more than four 
dynamical coordinates, since not more than four independent ^-numbers 
can mutually commute. For that reason we have chosen to consider four 



8-5] The Differential Wave Equation 129 

dynamical coordinates in this section. The same general dynamical theory 
applies to any number of coordinates; but the problems involving four 
coordinates will be classed as one-body problems, and therefore come first 
in our order of treatment. 

Of the four dynamical coordinates one is singled out to be the proper 
time s, and its conjugate momentum W is called the hamiltonian.f The 
reason for this selection must lie outside the system itself; for the dynamical 
equations (8-52) are perfectly symmetrical. The distinctive property of s 
can only appear when we contemplate the system in relation to other 
systems. It is, as we have seen ( 8*1), the argument of the perturbations of 
and by other systems. The peculiarity that the system "goes on" in s, 
whereas it is merely extended in s l , s z , 3 , is explained if it is through s that 
changes in the system are linked to changes in the external world and 
therefore ultimately to the time sequence in consciousness. 

The principle that the separate physical systems into which we dissect 
the universe shall each have just one coordinate in common with the rest, 
is valuable as expressing the conceptions which are the basis of our nomen- 
clature. It is not so important that it should be fulfilled rigorously, since any 
supplementary coupling can be dealt with by perturbation methods. In 
adopting s as the unique link we assume an idealised standard environment 
of the system, such that any change in the environment produces effects 
which occur simultaneously in all parts of the system according to the time 
reckoning s. In special cases we may have to treat an environment which 
deviates markedly from this standard. If light waves are falling on an atom 
in a particular direction, we should take account of the fact that the per- 
turbation travels across the atom with the velocity of light. It would seem 
therefore that the distinction between s, s l9 s^ 9 $ 3 is a matter of degree; and 
that all four coordinates afford potential linkages with external systems of 
appropriate character, though only one is called into play by the standard 
environment which our equations presuppose. 

To sum up: it is idle to treat in our equations a system supposed to have 
no interaction with its environment, since the interaction is the only thing 
about the system which concerns observational physics. On the other hand, 
it is not necessary to go to the other extreme and treat a system with an 
environment of the most general kind that can occur in nature. Just as we 
begin by studying the simplest systems, so we begin by studying the 
simplest form of environment, capable of introducing only the simplest 
type of perturbation. Under these conditions one coordinate 8 plays a 
unique role, and becomes distinguished from s l9 s 2 > *s- 

t In our nomenclature. On all points which concern the relations of s and t our Out- 
look differs so much from the current theory that comparison of nomenclature is scarcely 
possible. 



130 Wave-tensor Calculus [8-6 

8-6. The Differential Wave Equation for a Strain Vector. 

The tetrad of matrices in the wave equations (8-261) or (8-262) may be taken 
to be either E l9 E^ E 3 , J0 4 or J5 16 , jE? 25 , J5 36 , j 46 . The choice is not entirely 
a matter of indifference, because the two tetrads lead to different reality 
conditions. When the p^ are algebraic, the former is the appropriate tetrad 
for a (four-dimensional) vector density 0^*> an d the latter for a space 
vector 0x*, by (6-51). Prom considerations of continuity the same distinc- 
tion must hold for non-algebraic wave functions. 

Accordingly the wave equations without electromagnetic field for a 
space vector 0x* are 



- < 8 - 6i2 > 

Let <* = ix*^45 > so tha ^ 8 = W* is the associated strain vector. Substituting 
in (8-612) we obtain 



<>' (8-62) 
which is equivalent to 

^ (8-631) 



since E^ is the only antisymmetrical matrix in (8-62). Multiplying (8-611) 
initially by E^ we obtain 

4+^4+g| + ** m )*- - < 8 ' 632 > 

Thus the covariant wave vectors 0, ^ are solutions of the same, differential 
equation. 

The equations for $ and ^ are, however, not the same when there is an 
electromagnetic field. In (8-611) and (8-612), m is replaced by 



so that in (8-62) and (8-632), E^m is replaced by 



But in passing from (8-62) to (8-631) the reversal of sign applies only to 
JS? 45 m. The equations for and ^, including electromagnetic terms, are most 
conveniently written 

=0 > (8-641) 



8-6] The Differential Wave Equation 131 

Remembering that i is contained implicitly in # 4 , * 4 and in the imaginary 
matrices JS7 14 , E^ , 1 34 , we see that if the sign of (8-642) is reversed it becomes 
the complex conjugate of (8-641). So that if (a? 1 # 2 #3* is a solution of 
(8-641), its complex conjugate is a solution of (8-642). There are, of course, 
other solutions of (8-642), and it is not necessary to suppose that the two 
wave vectors representing a particle are complex conjugates. 

The wave functions adopted by Diracf and used, I think, in all current 
treatises are $, <f>, not ^r, x- Even when ^ = 0, so that ty and <f> satisfy the 
same equation (8-631), they are taken to be different solutions representing 
waves travelling in opposite directions in four dimensions. This comes 
about because the momentum operators applying to them have been defined 
differently in (8-28); so that when the momentum has a given value (the 
same for *fi and </>) different functions are required. 

We shall call wave functions whose momentum operator is idfiXp + Kp 
wave vectors of index 1, and those whose momentum operator is id/dXp + Kp 
wave vectors of index 1. The former satisfy (8-641) and the latter (8-642). 
The definition will later be extended, so that a wave tensor is said to be of 
index n if its momentum operator is 

v (8>65) 



This applies to covariant, contravariant, initial or final wave tensors, 
d/dXp being changed to S/Sa?^ when the tensor is written initially. 

We have seen that <f> may be the complex conjugate of $. Dirac goes further 
and defines </> as the complex conjugate of 0. In the present theory there is 
no reason to impose this restriction, which is presumably a survival of the 
Hermitic conditions employed in Schrodinger's elementary theory. These 
are superseded by the reality conditions found in Chapter vi. For algebraic 
wave functions (8-285), ^r and ^ can be chosen independently from the 
infinitude of solutions of the elementary wave equation, and there is in 
general a similar independence of non-algebraic wave functions. 

Let us, however, consider for a moment the current theory which takes 
^ to be the complex conjugate of 0. The. full specification of the system is 
then contained in a single wave function 0; for we do not add anything to 
the specification by inventing a special symbol for the complex conjugate. 
The system might equally well be specified by <f>; then is merely a symbol 
for the complex conjugate of </>. But and <f> represent waves travelling in 
opposite directions in four dimensions. It may be asked, Which are the 
real waves, or are there waves in both directions ? The answer is that there 
are no real waves. I suppose that no one nowadays attributes objective 
existence to the waves described in wave mechanics. 

f Quantum Mechanics, 2nd ed., p. 255, equations (9) and (10). 



132 Wave-tensor Caleidus [8-6 

Since then the system is specified by a single wave vector 0, the most 
natural wave tensor of the second rank furnished by it would seem to be 
00* (or, with suffixes, 000), i.e. the outer square of 0. If 



where p^ is algebraic,f we have 

(-iO/a^+^W^W*. (8-66) 

Thus 00* is of index 2. By introducing the complex conjugate an alternative 
wave tensor 00* of index is obtained. Attention seems to have been devoted 
exclusively to the latter. It is of considerable importance; but we must not 
let it unduly divert attention from the primary wave tensor 00*. 

As already stated, we do not accept the limitation in Dirac's theory which 
reduces the specification of a particle to a single wave-vector function. The 
space vectors and strain vectors which comprise the ordinary vectors of 
physics are wave tensors of the second rank. We resolve these into pure 
constituents, which are factorisable into wave vectors. In general there is 
no reason to expect or require that the two factors shall be equal. I have, of 
course, no objection to the employment in quantum physics of wave tensors 
which are perfect squares, if these are appropriate to the problems which are 
studied as is sometimes the case. But to regard it as more than a casual 
adaptation creates an artificial gulf between quantum theory and relativity 
theory, since there is no such limitation in the latter. 

Thus a pure strain vector of index 2 will normally be the product of 
unequal factors /S 2 = 1 2 *. Exchanging one of these for its complex con- 
jugate, we obtain an associated strain vector of index 0, /S = 1 2 *. The 
latter is the strain vector we have been studying, but we shall now turn 
attention to S 2 . S 2 has the advantage that there is no need to factorise it. 
The momentum is given by the operator (8-65), used as in (8-66). Factorisa- 
tion is only needed for wave tensors of index 0. For them the operator (8-65) 
is indeterminate, and it is necessary to find a factor which is not of index 
so as to obtain the momentum. 

At present we treat only wave vectors of index 1. We may note, how- 
ever, that there is a possibility of extending the theory to wave vectors of 
any index n, integral or fractional, p^ being always given by (8-65). 

8*7. Application to Phase Space. 

In Chapter vn we have described a system by a strain vector which specifies 
simultaneously the configuration, the probability of the configuration, and 
the time. This cannot be the strain vector /S = 00*, which was introduced 
for the purpose of specifying the probability only, and is in fact independent 

t Or, more generally, if p^ is an ^-number containing only space-like matrices and there- 
fore commuting with 0. 



8-7] The Differential Wave Equation, 133 

of the time. The strain vector which generates phase space is the associated 
strain vector of index 2, viz. /Sg-^^j*. 

Consider a free particle in field-free space, so that the ordinary wave 
functions are = e iww ^ > *-*-*"*), (8-711) 

where ma =p 1 x 1 +p*x* +Pz^p Q t. 

Their product is the strain vector /S . Alternatively we specify the particle 
by wave functions <Ai> ^2> both of index 1, 

0i=e iw *(Wo> fc-d'-foJo. ( 8 ' 712 ) 

Their product is the strain vector 

S 2 = e 2 *m*OS a ) . ( 8 . 72 ) 

For a displacement <fc, the only change produced in & 2 is a change of algebraic 
phase dO l9 = 2mds. If displacement in time (proper time) is expressed in the 
same linear measure in phase space as in ordinary space, so that d0 16 = dsjR, 
we have 2rads = d$/-R, so that m=H2R (8-73) 

where R is the radius of the phase space. 

At first sight it is anomalous that the general displacement (dx l , dx 2 , d# 3 , 
dx) should be interpreted as change of time only, and not change of con- 
figuration. But the plane wave solutions (8-712) presuppose flat space-time. 
If they are used in curved physical space-time, they must be restricted to 
regions not too large to be treated as flat. This means that the region of 
phase space which they cover is not too large for the distinction between the 
different configurations to be neglected. The apparent discrepancy is thus 
due to the nature of the approximation assumed in plane wave solutions; 
although they formally distinguish configurations by coordinates, they 
suppose that the distinctions when expressed by matrices are so inconsider- 
able that they can be neglected. 

We have suggested ( 5-4) that wave vectors are introduced mainly to 
secure purity of the wave tensors, and that many if not all of the problems 
of quantum theory could be solved by using the wave tensors directly. 
Current theory gives a rather fictitious importance to the vector factors, 
because it recognises only wave tensors of index 0. These do not contain the 
factor e im *i and, since the energy m is one of the most important character* 
istics of a system, it is necessary to examine the factors in order to find it. 
This is avoided by the use of wave tensors of index 2, which contain the 
factor e* im *. There is then no need to have recourse to the wave vectors, at 
any rate so far as the calculation of m is concerned. 

The position may be summarised as follows. Certain properties of a system 
are naturally described by constant symbols, e.g. a steady distribution of 
probability or probability flux. When these are factorised, they are resolved 



134 Wave-tensor Calculus [8-7 

into components 0, </> whose time factors cancel one another. But, since the 
time factors cannot be supposed to exist solely for the purpose of cancelling 
one another, this is a tacit admission that the constant symbols do not 
comprise the whole data of the system, and that in a complete description 
time factors must appear. Chapter vii gives this more compendious descrip- 
tion in terms of strain vectors variable with the time; they are factorised 
into components whose time factors reinforce one another. 

The general dynamical theory of a system described by strain vectors of 
index 2 is analogous to that developed for space vectors in 8*4, 8*5. For 
a transformation q = e iw * 9 we have 

S' = ^'<*' = e iwd * . ^e iWda = e iwd8 Se iWd8 , (8-74) 

where W<f> = <t>*W. (8-75) 

The determination of W from W presents no difficulty, remembering that 
(djdx)<fr = <l>* (8/8x). From (8-74) we obtain the general dynamical equation 
for strain vectors 



(8-76) 

The transformation introduces a strain common to all the strain vectors of 
the system. It may be regarded as defining parallel strain in the same way 
that the transformation of the space vectors in 8-4 defines parallel dis- 
placement. The dynamical coordinate s measures a progressive parallel 
strain of the system. In elementary examples parallel strain is merely the 
internal aspect of what is externally regarded as parallel displacement of 
part of a system. 

8- 8 . The Electromagnetic Potentials . 

There exists an important transformation which leaves invariant the 
momentum vector p^ = i djdx^ + K^ . Let 

0' = e^iAe0 = A0, (8-81) 

where 16 is an algebraic function of the coordinates x^ . Then if 



we havef 
Hence 






Accordingly the transformation 0->0', K^-**/, defined by (8-81) and 
(8-82), leaves p^ unaltered. The electromagnetic field of force is also un- 
altered, because the addition of an arbitrary gradient to the potentials has 
no effect on the force. 

t It is understood that p^ here denotes the value (number or matrix) of the component 
of the momentum vector, not the operational expression -td/&*? M +K M . Thus p^ commutes 
with A. 



8-8] The Differential Wave Equation, 135 

The corresponding transformation of other wave tensors depends on the 
index n of the momentum operator ( i/n) 3/3a? M + K^ . The general law is 

f = AV, (8-83) 

where n is the index of ^ ( 8-6). Since the transformation is algebraic this 
applies to wave tensors of any rank (/S'=A n ), the index of the tensor 
being the sum of the indices of its factors. In particular a wave tensor of 
index is invariant. 

The foregoing transformation will be called a gauge transformation, 
because it is the adaptation to wave mechanics of Weyl's gauge trans- 
formation in relativity theory. We may regard the electromagnetic potential 
Kp as having been created by a non-integrable gauge transformation of 
neutral space-time. If in (8-82) we determine 16 so that SO^dx^ = 2^ , we 
have Kp = 0; that is to say, the gauge transformation removes the electro- 
magnetic field which may therefore be created by the inverse transforma- 
tion. But the equations dO^dx^ = 2^ , determining the transformation, are 
non-integrable unless curl AC = 0. 

To justify the name "gauge transformation" we proceed as follows. If 
16 is imaginary, A is real, and the strain vector $ 2 which generates phase 
space is multiplied by a real algebraic factor A 2 . By 7-7 this represents a 
change of the probability of a range of configurations at the point considered, f 
But the probability is also given by ^*^dF, where ^r, ^* are the ordinary 
wave functions of indices 1 and - 1. J Usually changes of probability are 
expressed by changes of the modifying factor <[>*fa but in this transformation 
^*^r is invariant, since it is of index 0. The transformation therefore changes 
the measure of volume to dF' = A 2 dF. That is what is meant by a gauge 
transformation a change of measure of volume (implying a change of the 
standard of length]_without alteration of the coordinates. In terms of 
coordinates dV=*V gjk, dV'^V^-g' .dr, so that the transformation can 
also be expressed as V g' = A 2 V^- g. 

In Weyl's theory it was taken for granted that changes of electromagnetic 
potential correspond to real changes of gauge. Wave mechanics introduces 
an important amendment. By (8-82) real changes of the electromagnetic 
potential (fc lf * 2 , * 3 , ic ) correspond to real values of 18 , and hence to 
imaginary changes of gauge. The need for this amendment became obvious 
as soon as it was discovered in quantum theory that the significant com- 
bination is - i 3/3^ H- Kp , not 3/Sa^ 4- ^ . 

This amendment removes the only difficulties noticed in the unified 

t Note that this interpretation only holds if A is real. If A is complex (0 W real) the change 
affects the time coordinate and has no effect on the probability. 

J For simplicity we suppose that the transformation redistributes the probability 
without altering the total amount, so that it is not necessary to re-normalise after the 
transformation. 



136 Wave-tensor Cakulus [8-8 

gravitational-electromagnetic theory .| In the attempts to find a geometrical 
invariant representing the Action (loc. cit., pp. 230-3, 257), the difficulty has 
been that in the elementary invariants the total action and the electro- 
magnetic action F^ Fw occur in the combination G + F^ v Ft* v . In particular, 
the generalised volume V(- I *^wl)> which has since been brought into 
prominence by its use in the Born-Infield theory, reduces to this combina- 
tion (loc. cit., p. 233). But there seems to be no sense in adding an electro- 
magnetic action to a total action which already includes it. It is the differ- 
ence G-FpyFP", representing material or non-Maxwellian action, which 
should be represented by the elementary invariants. Devices for changing 
the sign, e.g. by alternating the suffixes in the invariant *G r /4r * (?"'*, were 
proposed, but were not very convincing. But the sign is rectified now that 
we realise that, owing to the identification of the electromagnetic potential 
with real instead of imaginary changes of gauge, IK^ has been substituted 
for Kp and iF^ for F^, throughout the field theory as originally given. 
Consequently F^F^ should have been -F^F^. 

With this amendment the field theory as set forth in Chapter vn of The 
Mathematical Theory of Melativity is acceptable today. The investigations 
in this book have a close connection with it at many points, and confirm it 
by elucidating the manner in which it forms the macroscopic counterpart of 
wave mechanics. 

After introducing gauge systems transformable at will, Weyl pointed out 
that there exists at every point of space-time a natural gauge furnished by 
the radius of spherical curvature; and he later reached the conclusion that 
our actual measures are made in terms of this gauge. This was extended 
by the writer who showed that a natural gauge, not only at every point but 
for measurement in every direction at that point, is provided by the con- 
tracted curvature tensor, and that the law of gravitation is the expression 
of the fact that it is to this gauge that our actual macroscopic measurements 
refer. 

By starting with no determinate gauge system, and thereby discovering 
the natural gauge instead of merely postulating it, Weyl had made a funda- 
mental advance. But, in a sense, his conclusion stultified his premises. The 
principles of physical measurement are bound up with the natural gauge; 
we cannot employ alternative gauges without giving to the words "length ", 
"volume", etc. meanings which they do not bear in physics. Gauge trans- 
formation had become one of those etymological transformations which 
too frequently mar theoretical discussions proclaiming the obvious truth 
that if you alter the meanings of words you may assert anything you like. 
In particular, one of the most attractive features of his electromagnetic 
theory had to be given up, viz. that the arbitrary gradient, which can be 
t Mathematical Theory of Relativity, Chapter vn. 



8-9] The Differential Wave Equation 137 

added to the electromagnetic potential without altering anything observ- 
able, represented the arbitrariness of gauge. Thus Weyl's two results (1) the 
discovery of variable gauge, which accounted for the existence of quantities 
which might be identified with electromagnetic potentials, and (2) his 
discovery of natural gauge, which leads ultimately to the explanation of the 
law of gravitation, seemed to be contradictory; and it was necessary to 
suppose that (2) superseded (1). But we can now accept them both, with the 
modification that the variability referred to in (1) is an "imaginary gauge 
transformation"; that is to say, it is not a change of the real part of log A 
which furnishes the standard for the measurement of lengths and distances, 
but of the imaginary part of log A, which (although called a gauge trans- 
formation by analogy) does not affect the reckoning of length. 

In our present development natural gauge is used from the beginning, 
because displacement first arises as an angular quantity (angle of a trans- 
formation) which is the ratio of the linear displacement to the radius of 
curvature. Thus we do not encounter the preliminary ambiguity which 
leaves the measure of the displacement indeterminate until the radius of 
curvature is brought in as standard.f There is no provision in our theory for 
real change of gauge for using any other standard. The reality conditions 
for rotations restrict the transformation (8-81) to imaginary gauge trans- 
formations, i.e. real changes of the phase angle 16 . 

8- 9 . Non-integrable Transformations . 

In a general way we can trace the origin of the non-integrable gauge trans- 
formation which creates an electromagnetic field. A non-integrable trans- 
formation arises when we contemplate a field of transformation composed 
of transformations which do not commute. For example, the transformation 
^'==e^aid*rM*2jfcjj^ is non-integrable. If we apply it to a square circuit 
composed of successive displacements (dx lt 0), (0, dx 2 ) 9 (dx l9 0), (0, dx 2 ) 9 
we obtain as far as the second order 

0' = (1 - E 2 * 2 dx 2 - frfdvf) (1 - 



which reduces to $' = (1 - 2E l E^c^c^dx^dx^) $. 

Thus the result of taking round the circuit is to transform it to 

^Be-t^s****!**!^. (8-91) 

This, however, does not immediately solve the problem of the creation of an 
electromagnetic field, which depends on a similarly non-integrable algebraic 
transformation. 

t We are able to start in this way because we treat a very simple uniform space-time, 
whereas the field theory is concerned with the origin of the natural gauge system in irregular 
macroscopic space-time. 



138 Wave-tensor Calculus [8-9 

The field K^ is due to systems extraneous to the particle or system S to 
which the wave vector belongs. As explained at the end of 2*9 the 
external particles have their own symbolic frames F^ etc., which commute 
with the ^-symbols of the system 8. The effect of recognising these external 
particles will be to introduce into our calculus a large number of additional 
symbols which commute with the E^ but not in general with one another; 
so that a much wider variety of transformations q can be contemplated. 

We have hitherto ignored the external particles and the transformations 
representing relative displacement of them, because the elementary equa- 
tions suppose S to be in a standard environment, namely neutral space- 
time. But an electromagnetic field presupposes a non-uniform environment. 
A change of position dx^ is not merely a transformation from one point to an 
equivalent point of space-time (4-4); it involves also an intrinsically 
different environment of 8. Thus the displacement dx^ will involve a 
supplementary transformation, representing the change of environment, 
which we may take to be of the form 

0' = exp{F 1 a 1 ^ 1 + 7 2 a 2 da: 2 + r3a3^3 + 7 4 a 4 da: 4 }.0, (8-92) 

where the Y^ are composed of symbols belonging to the extraneous systems, 
and therefore commuting with the E^. We do not suppose that the Y^ are 
anticommuting symbols of a complete set; they will usually be complicated 
symbolic expressions. But provided that they imperfectly commute (as 
normally happens with complicated symbolic expressions), the trans- 
formation (8-92) will be non-integrable; and 0, after being taken round a 
circuit, will not return to its original value but will undergo a F rotation of 
some kind. But since the F-symbol of the rotation commutes with all the 
Ep , it will be indistinguishable from an algebraic transformation; and it will 
count as an algebraic transformation so far as the Enframe is concerned. 

Thus the effect of irregularity of distribution of the surrounding protons 
and electrons, which might be particularised with almost an infinitude of 
detail by introducing their own symbolic frames, is reduced to a non- 
integrable algebraic transformation of the vectors of the -E-frame. This 
transformation represents the difference between the standard environment 
of neutral space-time and the modified environment a difference which 
is recognised as the macroscopic electromagnetic field due to the specialised 
distribution of the external charges. As we have seen in 8-8, a field of non- 
integrable algebraic transformation is equivalent to the insertion of electro- 
magnetic potentials K^ in the momentum vector. 

We have supposed that the supplementary transformation (8-92) contains 
only the symbols belonging to the external systems. Would it not be more 
natural to suppose that it contains combinations of these external symbols 
with the symbols of S, e.g. E^l In that case the non-integrable trans- 



8-9] The Differential Wave Equation 139 

formation will not be algebraic in the .E-frame, and the field cannot be 
represented by a potential vector with algebraic components K^ . I agree 
that it would be more natural. But the question for us is, not what actually 
happens, but what is supposed to happen in the ideal problems to which 
Dirac's equation (8-262) is applied. Actually a charged particle polarises 
the surrounding distribution of electric charges. When it is displaced the 
potential due to surrounding charges is altered by their changed polarisation. 
This effect (Debye-Hiickel effect) is of great practical importance. But it is 
not included in Dirac's equation, which postulates that the field is due to a 
rigid distribution of charge. If Dirac's equation is applied to an electron in 
a field in which the Debye-Hiickel effect is large, it gives an incorrect value 
of the energy. 

Thus in treating the origin of the K^, i.e. of the electromagnetic terms in 
Dirac's equation or equivalently the electromagnetic terms in the momen- 
tum and energy, we must adhere to the same idealised conditions. The 
postulate is that the system 8 itself has no share, direct or indirect, in 
determining K^. We must therefore omit the terms, if any, which are not 
invariant for rotations of the particle, substitution of particles of opposite 
signf or opposite spin, etc. The terms admitted therefore correspond to a 
purely algebraic transformation. There is no need for us to show that the 
omitted terms are small in practical problems; very often they are not. 

The term K^ in the wave equation (8-262) is essentially a microscopic 
electromagnetic potential. In microscopic problems the field due to one or 
more individual particles requires a more complex specification by means of 
multiple matrices. The criterion is that, if the distribution of particles 
producing the field can be treated as rigid, (8-262) suffices. The foregoing 
discussion makes it clear that the field due to the particle itself is not to be 
included in K^. Neglect of this condition has led to the occurrence of an 
infinite self-energy of the particle in certain theories. 

The internal wave equation for the hydrogen atom, adopted in (9-221), 
provider* an exception to the rule that K^ is a macroscopic potential. The 
equation is of the form (8-262) notwithstanding that the electromagnetic 
field is due to a single particle (the proton). This is because the problem is 
transformed by the use of relative coordinates into the motion of a particle 
in a rigid field. It must be emphasised that this is a quite exceptional use 
of *, made possible by the simplicity of the problem, and that the micro- 
scopic interactions of particles cannot usually be represented by a field of 
this form. 

t There is an apparent change of sign of the electromagnetic terms in the wave equation 
when a proton is substituted for an electron; but what has really happened is that the 
electromagnetic terms are unaltered, and all the other terms have changed sign. 



CHAPTER IX 
THE HYDROGEN ATOM 

9*1. Steady States. 

Before tackling a practical problem, it is appropriate to recapitulate and 
systematise certain ideas which have appeared in a scattered way in previous 
chapters. 

In the practical application of wave mechanics the central problem is the 
search for systems which shall be dynamically steady. The phrase ' c dynamic- 
ally steady" requires amplification. 

There is an almost inevitable ambiguity in the use of the words ' ' electron ' ' 
and "proton " in the new physics. We say that (a) an electron is no longer a 
particle but a wave, and (6) that the waves specify the probability dis- 
tribution of an electron. Thus the term is applied both to the distribution and 
to that which is distributed. For definiteness let us call that which occupies 
any point of the distribution an electron-point. We consider then electron- 
points distributed over a domain of geometrical coordinates x^ . A displace- 
ment dXp is a displacement of the electron-point that is contemplated; no 
dynamical conception is attached to the displacement; it is a transfer of our 
attention from one electron-point to another. But we can contemplate also 
a bodily displacement of the whole distribution; such bodily displacement 
is described as a change ds^ of a dynamical coordinate s^ . Here again the 
displacement may be regarded primarily as a transfer of attention from 
one electron-distribution to another, instead of from one electron-point to 
another. But when, by habit, we introduce dynamical conceptions they are 
attached to the displacements ds^, not to dx^. The dynamical electron 
the moving entity is the probability distribution, and its mode of dis- 
placement is wave propagation. 

We need not confine attention exclusively to bodily translation or rotation 
of the distribution. We can consider more general sequences of distributions. 
The general method of specifying a sequence of distributions is by a trans- 
formation q = e iw *\ then a displacement ds signifies the change of distribution 
which is produced by applying the transformation e iwds to the vectors 
describing the distribution. If W is an J5-number, this is a relativity rotation 
of the space vectors defining the distribution, and is therefore a displacement 
without intrinsic change. The corresponding coordinate 8 will be called a 
simple dynamical coordinate. By allowing W to include differential oper- 
ators, we obtain a more general type of displacement including deformation, 
and define a correspondingly generalised dynamical coordinate. Each simple 
dynamical coordinate is closely related to (and frequently confused with) 



9-1] The Hydrogen Atom 141 

a geometrical coordinate x^ or 0^, viz. that defining the direction of the 
bodily displacement or rotation of the distribution; but the generalised 
dynamical coordinates have no geometrical counterparts. In practice, 
however, they often have approximate counterparts; for a generalised 
coordinate usually appears as a slight modification or adaptation of a simple 
coordinate. For example, a free electron possesses (simultaneously) four 
simple dynamical coordinates representing bodily displacement of its 
distribution in four antiperpendicular directions x l9 # 23 , # 45 , a? 16 ; when the 
electron is in the electromagnetic field of a nucleus, we have to find four 
generalised dynamical coordinates to replace these. 

Let us consider a system with four dynamical coordinates s^. In what 
circumstances should we describe the fourfold sequence of distributions as 
steady 1 It would certainly be considered steady if the distributions were 
all intrinsically similar. But that is unnecessarily stringent, since we cannot 
make exhaustive observations of every detail of the distribution. The 
minimum condition is that some recognisable characteristic of the distribu- 
tion shall be steady, i.e. constant over the domain of dynamical coordinates 
8p . Since the observable characteristics (physical vectors) are space vectors, 
we require that a complete space vector J determined by the distribution 
shall be constant over the domain s^ . That is to say, J must be invariant 
for the transformations q = e iw * 8 *. 

We take J to be factorisable. It would be possible to obtain a steady state 
by compounding two pure states neither of which is steady. But the com- 
bination is not of practical importance unless there is security that the two 
states remain superposed with the same relative probability factors when 
external perturbations are admitted. The argument runs: the steady states 
which we wish to discover are those which behave as units under external 
perturbations. Unitary character, i.e. purity, is expressed symbolically by 
a spectral operator. Therefore to bring our symbolism into line with the 
physical conditions to which it is applied, we must represent the unit states 
by spectral operators. The latter are idempotent symbols. We consider in 
particular the idempotent space vector as the simplest element in the 
symbolism that is thrust upon us. We regard the simple elements of the 
symbolism and their physical counterparts, not as hypothetically "exist- 
ing ", but as idealisations which owe their importance to the fact that any- 
thing more complicated can be, and commonly will be, analysed into these 
simple elements. 

By 8-5 a sufficient condition that J shall be constant over the domain 
s, ! , 8 2 , s 3 is that its factors ^r, x* shall be common eigensymbols of W, U^ , 
U 2 , U$. It is easily seen that this condition is also necessary. 

In practice we assume that W, U l9 U 2 , U 3 commute. It is not true that 
operators which have a common eigensymbol necessarily commute ( 3-7 (/)) 



142 Wave-tensor Calculus [9-1 

but if W, #i, U 2 , J7 3 do not commute they will not be constant over the 
domain of s, s l9 s 29 s^ 9 their derivatives being given by (8-52). Transforma- 
tions in which the operational forms are functions of the dynamical co- 
ordinates, e.g. VP(*i,*i,*a), are not considered in wave mechanics at 
present. They bear the same kind of relation to constant transformations 
that general relativity transformations bear to those of special relativity 
theory. The domain of such transformations will have a curvature embodying 
the non-commutability of the rotations in it. Whether it would be profitable 
to pursue the study of non-commuting symbols with a common eigensymbol, 
I cannot say. But it may be worth noticing that existing methods of search 
for steady systems, i.e. distributions with a recognisable characteristic 
which is constant over a multi-dimensional domain, are not necessarily 
exhaustive; and it is just possible that steady states, which may have some 
physical importance, have escaped our analysis. 

Limiting ourselves accordingly to constant transformations, the problem 
of finding dynamically steady states resolves itself into the finding of four 
commuting symbols, or whatever number of symbols may be appropriate to 
the kind of system investigated. The symbols represent constant cha- 
racteristics of the system; but they are generally portions of the constant 
space vector J, and do not imply any constancy of the system additional 
to that originally postulated. For example, when the operators are E- 
symbols, W + U^U^U^E^ + E^E^ + E^^J. 

We must next try to understand why these ' ' steady states ' ' are important . 
Only one of the four coordinates is conceived as time displacement. Our 
dynamical picture of a system pursuing a trajectory in the domain of 
s 9 SJL, s a , 5 3 does not suggest any reason why J should be required to be 
constant in directions transverse to the trajectory. The importance of the 
latter condition is that it introduces the maximum degeneracy into statis- 
tical enumerations. In accepting J as the criterion of steadiness, we 
implicitly decide to count all configurations which have the same J as one 
configuration. Thus the probability occupying the whole of S-space counts 
as the probability of one configuration. Our picture of a configuration 
as an isolated point in /S-space pursuing a trajectory does not apply; it is 
a whole continuum or wave front that "travels". Moreover, when (as in 
quantised systems) the four-dimensional /S-space exists only for discrete 
values of J 9 and intermediate values of J occupy loci of three or fewer 
dimensions, the discrete values have infinitely greater probability than the 
intermediate values. 

An /S-space formed by generalised dynamical coordinates is not on quite 
the same footing as an /S-space formed by simple dynamical coordinates. 
The difference is that the generalised displacement is a transformation 
peculiar to the system, and is not applicable to its idealised environment. 



9-2] The Hydrogen Atom 143 

Now a system without an environment is unthinkable; and it is no use 
displacing it all over the S -space if it cannot take with it the environment 
which its structure demands. That was the early mistake of relativity 
theory, which applied transformations to the differential equations but 
omitted to provide for their application to the boundary conditions. The 
standard environment is uniform, i.e. spherical, neutral space-time. This is 
conceived as permanent; so that the only transformations admitted are 
those which transform it into itself, namely the kinematical ^/-rotations. 
Hence, in general, if we apply generalised W transformations to a system, it 
will no longer fit the boundary conditions where it merges into standard 
space-time. Physically we should say that the new configuration requires 
a pressure or an electromagnetic field to maintain it. 

It might seem from this that the importance of generalised steady states 
is fictitious. But we have to remember that the standard environment is a 
simplification of the actual environment. The practical physicist is not 
concerned with a hydrogen atom existing alone in uniform spherical space. 
He deals with hydrogen atoms surrounded by other atoms or ions, or in 
fields of radiation. We admit that it would be useless to consider generalised 
displacements of a system which could not be applied to its environment; 
but this is discounted by the fact that we cannot say beforehand what the 
exact environment will be, and what type of displacement it will admit. 
We therefore investigate the most general steady states, since they may 
become realisable if the environment is appropriate. 

To conform to our observational knowledge there should be just one 
steady state of a hydrogen atom in the standard environment, namely the 
ground state. The other states can be realised if the environment is such as 
is capable of "exciting" the atom in particular if it contains a field of 
radiation. 

We proceed to an investigation of the steady states of a hydrogen atom. 
This belongs to quantum theory proper, and strictly speaking is outside our 
territory. But it is necessary for liaison purposes to follow through in our own 
notation, and from our own point of view, one practical problem which 
introduces the leading ideas of quantum theory. Unexpectedly the in- 
vestigation has proved to be vitally important, because it reveals an in- 
consistency of a factor 2 (for which, I think, the quantum physicists must be 
held responsible) which led to an error in the numerical results found in 
earlier versions of the present theory. 

9-2. The Commuting Operators for a Hydrogen Atom. 

The conditions for a steady internal state of a hydrogen atom were obtained 
approximately by Schrodinger as a differential equation of the second order. 
This is now superseded by Dirac's exact treatment. It is commonly said that 



144 Wave-tensor Calculus [9-2 

Dirac replaced Schrodinger's second order equation by a first order equation 
(W m)<Jj = 0. I regard this as a misconception. The modern equivalent of 
Schrodinger's equation is the set of partial differential equations 3 W/ds^ = 0, 
30i/8* M = O f etc., which secure that certain characters remain steady for 
fourfold displacement. Since W, U l9 U 2 , U$ contain differential operators, 
these equations are of the second order. The introduction of ^-matrices has 
not affected the fundamental conclusion that the condition that ifi shall 
represent a steady state is expressed by differential equations of the second 
order. 

By the dynamical equations (8-52) the equation 3W r /3 1 = is equivalent 
to 4 W WUi = 0. In this way the original set of equations is reduced to 

the condition 

W, U 19 U 2 , C7 3 mutually commute. (9-21) 

We may regard (9-21) as a first integral of the second order equations. 

The way in which Dirac's wave equation enters into the problem is that it 
fixes the analytical form of one of the symbols W in (9-21); and hence it is 
the starting point for determining three other analytical forms which com- 
mute with it and with one another. 

We have shown in 9-1 that $ must be a common eigensymbol of W, U l9 
U 2 , J7 3 ; so that, if ra is the eigenvalue of W, we have (TF-ra)^ = 0. This 
equation is provided by (8-261), which we found as a condition for the con- 
servation of probability . We could not at that time show that m was constant 
in time; but this now follows from the dynamical equations, because W is 
constant. We have still to show that m is the same for different states of the 
system. It will be proved in Chapter xn that (for particles represented by 
simple wave tensors) m has one of two absolutely determined values, corre- 
sponding respectively to electrons and protons. Using this result in anti- 
cipation, we shall take m to be a constant of nature. 

The eigenvalue m of W is imposed from outside; there are no limitations 
on the eigenvalues of U l9 U 2 , Z7 3 other than those which will be found in the 
course of the investigation. This difference is due to the fact that s is singled 
out as the connecting link with extraneous systems ( 8-5). 

We now consider the wave equation (8-262) for an electron in an electro- 
static field of positive potential proportional to 1/r, so that 



^-(0,0,0, -t/r), f^- 
Then writing (8-262) in the form (JF-w)^ = 0, we have 



We seek three other operators, linear in 9/3^, which commute with W and 



9-3] The Hydrogen Atom 145 

with one another. It is easily verified that the following satisfy this condition : 



~ 1 ' (9>222) 
< 9>223) 

^=a=-- (9>224) 

We denote the eigenvalues of U , Z7 2 , U z , JF by /A, u, 9 m, and our problem is 
to find the relations which must exist between them, in order that the com- 
mon eigensymbol ^ may satisfy appropriate boundary conditions. More par- 
ticularly we wish to determine c, which is the energy conjugate to , in terms 
of w, /A, u. The method of solution here followed is mainly due to Temple.f 
We have enunciated the problem in an abstract way, without mentioning 
the hydrogen atom. But (9*221) is the hamiltonian of the hydrogen atom 
adopted in Dirac's theory, and we shall provisionally assume that that is 
the physical application. There is no obvious reason to expect that the wave 
equation (8-262), in which the K^ are potentials of a macroscopic electric 
field ( 8*9), will be adequate in the interior of an atom; so that for the time 
being the term E^OL/T in (9-221) is only justified empirically. But in Chapter 
xv we shall find that (9-221) is the exact hamiltonian for a system of two 
elementary particles and we shall determine the theoretical value of the 
constant a. 

9*3. Solution of the Equations. 

Let E r ^(E l x l + E^x^E, A x^lr (9-311) 

so that E r 2 = 1, and E r anticommutes with E. By direct multiplication 






r 3 a^ ~ Xl te 

tj^&i-l. (9-312) 



Hence by (9-221) 

(9-313) 



Multiplying this by the common eigensymbol ^, the operators W, U i9 
reduce to their eigenvalues m> p, *, so that 



" ai (9 ' 321) 

t Proc. Roy. Soc. A, 127, 349 (1930). 



146 Wave-tensor Calculus [9-3 

Or, writing ^ = ^01, 

= Q. (9-322) 



T ^4- "El Jf Jjl _ _|_ fl * If /Wl /2 A ~Ifl mm U' B' Jk* /AOQ1\ 

XJ\Jv jF "- "" p ^A ^ "i ** r t"j ^y sag p py < jLv * * fwm "f^A QC l7 OJA) 

so that the equation is da> / v Q\ 

-j- 1 jj I" i 1^ ss y, i y*oo^& ) 

c/y \ r / 

By (9-331) -F 2 =/ a , (? a =8' a , FG+GF= -2*e, (9-333) 

where /=(m a - a )*, gr^-a 8 )*. (9-334) 

To solve (9-332) we make the algebraic substitution 

r=x/2f, a>=e-* x $, (9-341) 
and take and x as our new variables. The equation becomes 



In an ordinary algebraic equation this transformation would have re- 
moved the last term; but in the symbolic equation it leaves a singular 
coefficient J (F/f 1) instead of a zero coefficient. Assume a solution in series 



(9-351) 
o 

where the C n may be non-algebraic. In order that the integral of the three- 
dimensional density ^(f>*r 2 dr may be finite for a region enclosing the 
origin, 0r and </>*r must be finite or else tend to infinity less rapidly than r~*. 
Hence co is finite or diverges less rapidly than r~* f so that 

p>-\. (9-352) 

Further, in order that the same integral may converge when the region 
extends to infinity, the series must terminate at some finite value of n, say 
n = n'. (It can be proved that, unless the series terminates, it diverges 
when #->oo.) 

Substituting (9-351) in (9-342) and equating coefficients of x n +-\ we 
obtain the recurrence relation 

(*+p + Q}0 % --l(PU-l)Ort. (9-361) 

Setting n = 0, (p + G) <7 = 0. So that 6 f is an eigensymbol of G, and p is an 
eigenvalue of G. Hence, by (9-333), p = g. We shall find later that there is 
no possible value of g between and , so that by (9-352) p =g. 

Setting w=w' + l, we obtain %(F/f l)C n , = 0; so that C n . is an eigen- 
symbol of F, the eigenvalue being/. 

Multiply (9-361) by (F/f+ 1); since jP^/ 2 , we have 

(F/f+l)(n+g + G)C n = 0. (9-362) 

Hence, by (9-333), 

0. (9-363) 



9-4] The Hydrogen Atom 147 

Now set n = n'. Since FC n '=fC n *, we obtain 

rc' + gr-<X//=0. (9-371) 

Or, by (9-334), c 



a 



This is Sommerfeld's formula for the energy . Here n r is a positive integer 
or zero. It remains to determine the possible values of p. 

9-4. The Eigenvalues of U^ U 2 . 

Take spherical polar coordinates r, 0, <f>, so that </> is the aziniuthal angle in 
the plane # 2 a? 3 . Then U 2 = i 3/3^4- \iE^\ and, since u is its eigenvalue, 

(Ut-u)$ = (-idfi<l> + \iEw-u)$ = Q. (9-411) 

Let x = e^' tt +*^3>^. (9-412) 

Then fJ 

= by (9-411). 

Accordingly x & constant for change of ^; and since (assumed to be 
single-valued) is unaltered when (f> is increased by 2?r, we have 



Or, since e* ^ = 1 = e* w , 

^r = e-^ + *> 2w ^. (9-413) 

Therefore w + is a positive or negative integer. The eigenvalues of Z7 2 are> 
therefore the half odd integers positive and negative. 
To find the possible values of JLI, we write 

a>i = xJltot-xJlfai+ JjB MJ etc. (9-421) 

Then it is easily verified that 

<>i<>2 ctf a a> 1 = co 8 , etc. (9-422) 

Hence 

<o 1 (co 2 + to 3 ) = (a> 2 + ia>3)(a> 1 --i), (9-431) 

a>! (a> 2 - ia> 3 ) = (a> 2 - ia> 3 ) (^ + i), (9-432) 

(a> 2 + ia> 3 ) (a> 2 i<o 3 ) = w 2 2 + a> 3 2 icoi , (9-433) 

(a> 2 - ia> 3 ) (a> 2 + ia> 3 ) == a> 2 2 + co 3 2 + ia> L . (9-434) 

By (9-222) and (9-223) 

tJ? 4 {7 1 ^ i8 ai 1 + lf8 1 cii t + JB u cii, + i 9 02 = ^!. (9-441) 

Since J? 4 commutes with U l9 we obtain by squaring the first expression, 



using (9-422). Hence 

(9-442) 



10-2 



148 Wave-tensor Calculus [9-4 

and, by (9-434), 



(9-443) 

Our solution ^ is an eigensymbol of U 19 U 2 , but not of E . We can, however, 
obtain a common eigensymbol xo of the three commuting symbols U l9 U 29 E^ 9 
with eigenvalues /*, M, - i, by the method of 3-7 (e) 9 namely #0 = (E^ - i) ^, 
since this gives (E^ + i) Xo = - Hence by (9-442) and (9-443) 

(9-451) 



We introduce a series of symbols defined by the recurrence relation 

Xr = K + *w 8 ) fr-i ' X-r * ( Wa ~~ ia) *) x - f +* ' 
r being positive. Multiplying (9-452) initially by (w 2 + iw 3 ), it becomes 

(co 2 4.fco 3 )(co a -ia> 3 ) Xl = {~(,,--l) 2 + (i + ^} Xl . (9-471) 
Also 

ico! xi = io*! (o> 2 + io> 3 ) xo = i (o> 2 + iw 3 ) (c^! - i) Xo 

by (9-431). Since U 2 =ia) l9 this gives 

^2Xi = K + ^3)(^+l)Xo = (^+l)Xi. (9-472) 

Adding (9-471) and (9-472) and using (9-433) 



2 } Xx- (9-473) 

By (9-472) the substitution of xi for XQ changes the eigenvalue of U 2 from 
u to u l = u+ 1; and, comparing (9-473) with (9-451), we see that the eigen- 
value of (w 2 2 + ci> 3 2 ) retains the same form with u changed to % . 
Proceeding step by step, we find that for x r > the eigenvalues of U 2 and 



Since U 2 =ia) l9 the eigenvalues of w^ are negative. Similarly the eigen- 
values of w 2 2 and o> 3 2 are all negative. If this implies that the eigenvalues of 
co 2 2 + o> 3 2 are all negative,! there is (for given /x) an upper limit to the value 
of u r 2 in (9-474). Hence the series of symbols XT must terminate with symbols 
Xfc> X-fc' sothat 

Xk+i = (^2 + ^3) Xk = 0, X-*'-i = (o> 2 - ico 3 ) x-tf = 0. (9-481) 
Adapting (9-452) to Xk> ^ e condition gives 



We find similarly 0=-(fi-l) 2 +(-i + u^) 2 9 so that 

( / i-l) 2 = (^ + fc + J) 2 = (^~&'-i) 2 . (9-482) 

Hence, for a fixed value of /*, the possible values of u range between 
| ^_ 1 1 - an d _ | JX- 1 1 + , the former limit corresponding to & = and 
the latter to *' = 0. 

t The legitimacy of this inference is examined later. 



9-5] The Hydrogen Atom 149 

For our purpose the pertinent result is that by (9-482), /A is a positive or 
negative integer; for we have already seen that u + $ is a positive or negative 
integer. We are therefore able to calculate the energies e of all possible 
steady states by giving n* and ft integral values in (9-732). Zero value of ^ is 
excluded because it would make the term VO* 2 - <* 2 ) imaginary; zero value 
of n' is not excluded. 

The assumption made above that o) 2 2 4- o> 3 2 has only negative eigenvalues 
requires consideration. I do not think there is a general law for non-com- 
muting symbols that if the eigenvalues of X and Y are all negative the 
eigenvalues of X + Y are all negative. But here we contemplate a restricted 
type of eigensymbol i/t. In one form or another conditions must be introduced 
which make the expectation value with respect to iff intermediate between 
the greatest and least eigenvalues. As this part of the theory of eigen- 
functions in wave mechanics does not concern us very closely, we shall not 
enter upon it. Accepting such conditions, the expectation values of c^ 2 and 
o> 2 2 will always be negative, and therefore the expectation value of w^ -f o> 2 2 
will always be negative. The eigenvalues, being particular cases of expecta- 
tion values, will therefore be negative. 

9- 5 . Metastable States . 

According to the theory which will be developed in Chapter xv, our adopted 
wave equation for the hydrogen atom is exact] that is to say, the interaction 
of the proton and electron is precisely expressed by a potential * 4 = icK.fr. 
There is no failure of the formula however small r may be. We have found 
exact solutions of the exact equations; and they agree with the well-known 
series of quantum states of the hydrogen atom. But we seem to have proved 
too much ! Observationally these states are only imperfectly steady; even 
if the atom is undisturbed, they do not endure indefinitely. Thus a state 
which satisfies the exact theoretical conditions for a steady state is found 
observationally to be imperfectly steady. 

The explanation is that in practical applications we have to take into 
consideration the environment as well as the atom itself (p. 143). We have 
found the exact solutions of the differential equations; but whether a par- 
ticular solution has an exact counterpart in nature depends on whether the 
boundary conditions which it demands are forthcoming. The boundary 
conditions of a "state " are difficult to visualise. Our general outlook is that 
whatever exists outside | the state is to be treated as a possible source of 
perturbations; an environment is therefore regarded as conformable to the 
boundary conditions of the state if it causes no perturbation of the state. 

Light is thrown on this subject by distinguishing between the algebraic 
and the non-algebraic wave functions of a hydrogen atom. The algebraic 
t In the sense of not belonging to the state, not necessarily exterior to it in space. 



150 Wave-tensor Calculus [9-5 

solutions of (9*332) are easily found. If CD is an algebraic wave function, the 
term d/dr in 



reduces to an algebraic function of r, so that o> must be an eigensymbol of 
F + O/r for all values of r . Therefore co is an eigensymbol of F and G. Then 
by 3-7 (/), FQ- OF is singular (or zero). By (9-331) 

$(FG-GF) = iE r n-E r Etmp + iEim<x,, (9-51) 

so that J ( FG - GF) 2 = c 2 /* 2 - m V + w 2 a 2 , (9-52) 

since E r , JS? 4 , J5? rJ E 4 anticommute. The right-hand side must vanish, because 
FG GF can have no reciprocal. We have therefore 

2 /(w 2 - 2 ) = O* 2 - oc 2 )/a 2 . (9-53) 

Comparing with (9-372), we see that the algebraic wave functions correspond 
to n' = 0. In this case the series (9-361) reduces to its first term, and the wave 
function is 



(9 . 54) 

It is well known that the states given by n' = are the metastable states. 
Accordingly the distinction between the metastable and the unstable states 
is that the former have algebraic wave functions and the latter non-algebraic 
wave functions. 

It appears therefore that, in order to satisfy the boundary conditions 
furnished by the standard environment uniform neutral space-time the 
wave function must be algebraic. Non-algebraic wave functions do not 
precisely satisfy the boundary conditions; but the discrepancy can be 
treated as a perturbation. The spontaneous transitions from these states 
to lower states are attributable to the perturbations which represent this 
discrepancy. Even the algebraic wave functions are not quite perfectly 
conformable to the boundary conditions, the only really permanent state 
being the ground state. 

9*6. Single -valued ness of the Internal Wave Function. 

We must now refer to the assumption italicised in the second paragraph of 
9-4. In determining the possible eigenvalues of U 2 and U l9 $is assumed to 
be a single-valued function of rectangular coordinates. But the assumption 
is untrue. Dirac's Lorentz invariant wave vector $ is necessarily a double- 
valued function of rectangular coordinates ( 4-7). Little attention seems to 
have been paid to this inconsistency; but it reveals a flaw in the foundations 
of the current theory of the hydrogen atom. 
We have to consider two alternatives: 

(1) The investigation in 9-4 must be amended so as to apply to double- 
valued instead of single-valued wave functions. 

(2) The eigensymbol is a single-valued function, and the investigation 
in 9-4 is correct ; but in this case cannot be a wave vector, and ( W - m) $ = 



9-6] The Hydrogen Atom 151 

cannot be Dirac's original Lorentz-invariant equation. Its origin therefore 
remains to be investigated. 

Our theory will be found to lead to the second alternative; but we may 
briefly consider the first. The result of taking $ to be double-valued is that, 
when the angle <f> is increased by 27r, may become either + or - $. We 
can then deduce that the eigenvalues of u are the integers and half -integers. 
No further change is made until we reach (9-482), which shows that the 
eigenvalues of p will also be the integers and half integers. We obtain 
therefore twice as many eigenvalues of p and of u as in the previous dis- 
cussion. The additional eigenstates do not correspond to observed states of 
the atom. 

Double-valuedness of the eigenfunction has been discussed by Temple.f 
It is pointed out that there would be a distinction between the integral and 
the half-integral eigenvalues, the latter being in a sense ineffective because 
a displacement has no matrix components corresponding to them. It is 
doubtful whether this ought to be regarded as an excuse for, or an objection 
to, a theory which employs the double-valued $. But the crux of the matter 
is the exclusion principle. The observational result is that when we consider 
a nucleus with a number of electrons, the electrons occupy the states given 
by integral values of /x and u + , and ignore the "ineffective" half-integral 
values. It would therefore be necessary to abandon the accepted form of the 
exclusion principle, and substitute a new principle according to which only 
a quarter of the cells, into which phase space is divided by the eigenfunctions, 
are allowed to be occupied. This would be a very drastic alteration of existing 
quantum theory. 

Following the second alternative, the only change required is to recognise 
that the eigenfunction of an internal state of the hydrogen atom (dis- 
tinguished as ^ H ) returns to its original value when the space axes are rotated 
through 27r, and therefore rotates twice as fast as the wave vector ^ (dis- 
tinguished as ifi D ) in Dirac's Lorentz-invariant wave equation. Thus fl 
transforms like a space vector or strain vector. We have seen that the in- 
ternal configurations of a system are specified by strain vectors, and we 
therefore take ifi H to be a strain vector. It must be of index 2, since a strain 
vector of index (representing a combination of waves travelling in opposite 
directions) would not have the unidirectional properties of H . 

Current theory has assumed that the momentum in practical units is 



t An Introduction to Quantum Theory, pp. 106, 131. His discussion (which is based on a 
treatment by Born and Jordan) refers to the Schrodinger scalar wave function. It is pointed 
out that (even in that case) ^ may not be a single-valued wave function, and that the usual 
inference as to the integral values of the angular quantum numbers is "decidedly precarious". 



152 Wave-tensor Calculus [9-6 

It is on this basis that the observational value of h is determined, and we 
must accept it as a definition of h. The momentum operator for a tensor of 
index 2 being iA/27r, the standard momentum operator for a wave vector 
of index 1 will be ih/Tr by (8-65). We have therefore 



The common assumption, that the momentum operator which applies to 
the wave function of a hydrogen atom applies also to the relativistic wave 
functions introduced in Dirac's Lorentz-invariant equation, is thus found 
to be untenable. The error of a factor 2 has escaped notice, because the 
Dirac wave functions are highly abstract (being referred to an unobservable 
geometrical frame) and are not directly concerned in comparisons of theory 
and observation; it is in connection with internal wave functions, which 
(as we have seen) are not Lorentz invariant ( 7- 1), that quantising conditions 
arise. But it is important that the factor should be set right in fundamental 
investigations of the connection between relativity theory and quantum 
theory. We have therefore to note that the theoretical unit of mass used in 
our fundamental investigations is such that A = TT, not 27r, and that the 
momentum operator for wave-vector functions is (in practical units) 

* * (9 ' 63) 

It is in accordance with the theory of phase space in Chapter vii that the 
configurations or states of the hydrogen atom should be discriminated by 
strain vectors of index 2 rather than by wave vectors. The theory of the 
energy levels of hydrogen thus falls into line with the general theory of the 
representation of internal configurations of systems. Looking at the matter 
from a physical point of view, the double-valuedness of \fi D reflects the abstract 
character of the analysis which introduces it. We can only observe relations 
between two systems or parts of a system, so that the minimum we can 
contemplate observationally is a double function *f> D .$ D '. Rotation of the 
axes through 2ir no longer introduces an ambiguity of sign; each factor 
becomes multiplied by e, and the product by e 27ri . When we consider 
relative motion, i.e. refer each particle to the other as origin, ^, ^/ 
become equal; ty D gives the distribution of the electron relative to the proton, 
and 02/ the distribution of the proton relative to the electron. In this case 
^H ^D-^D-t If we prefer to represent the state by a wave vector if/ D9 we 
must remember that ^= V^r H , and not ignore the ambiguity of sign 

| In this way wave tensors of index 2 which are perfect squares acquire a special 
prominence in practical problems. The wave tensors of index which correspond to them 
are Hermitic ( 8-6). This seems to be the origin of the Hermitic conditions, which have 
been applied too indiscriminately in current quantum theory. 



9-6] The Hydrogen Atom 153 

which is an essential characteristic of \fi' D . Taking the square root is, however, 
a gratuitous complication; and the actual investigation of the hydrogen 
atom is based on H . 

As a working rule, we note that normally the coefficient of the momentum 
operator will be ih/ir for absolute coordinates of a particle referred to a 
geometrical origin, and i A/2?r for its relative coordinates referred to another 
particle as origin. But it is impossible to lay down a universal rule, since the 
momentum operator depends on the index of the wave function adopted 
(8-65). 

Accordingly in practical units the special form for the hydrogen atom of 
the wave equation (8-262) of the double-valued wave function will be 

E *\- l *E ^-fiiUj-0, (9-64) 
dxj cr * J ^ D ^ ' 

since in practical units ^ = (0, 0, 0, fe 2 /cr). When, as in (9-221), the 
operand ^^ H = ^f D ^ D9 the coefficient A/TT must be replaced by h/2rr to give 
the same condition. It follows that the constant a in (9-221) is 

l/a = Ac/27re 2 . (9-65) 

Attention may be called to a further point on which we are unable to 
accept the current view. In (9-224), following the usual notation, we have 
denoted the coordinate conjugate to U B by t. We must now point out that t 
has nothing to do with time. Our equations refer to the internal states of the 
atom, and we have seen (7-1) that the only time- variable in an internal 
state is the dynamical or proper time s which is conjugate to W. If the atom 
is moving in an external frame the coordinate time in that frame is obviously 
not the t referred to in the equations. The dynamical coordinate conjugate to 
U 3 is of an altogether different nature, and will be identified in Chapter xv 
with the linearised permutation coordinate of the proton and electron. 



CHAPTER X 
DOUBLE WAVE VECTORS 

10* 1 . Multiplication of Probabilities. 

Probabilities are combined by multiplication. We have seen ( 7-7) that the 
configuration of a system and the probability of that configuration are speci- 
fied by a single symbol a strain vector. The multiplication of probabilities 
therefore involves a multiplication of the strain vectors; and the treatment 
of a combined system is based on the multiplication of the vectors descrip- 
tive of its separate parts. 

If the combined probability of two events is precisely the product of their 
separate probabilities, the two events are said to be independent. Similarly, 
we define independent systems to be such that the combined vector speci- 
fying probability and configuration is the product of the separate vectors. 
If in combining the systems the product is modified, the modification con- 
stitutes an interaction. For the present we consider combination without 
interaction. 

It is, of course, entirely opposed to our habit of thought to regard a system 
as the product of its parts rather than as the sum of its parts. Therefore we 
have a long way to travel before we can connect the combination of systems 
by multiplication with our ordinary outlook. 

Admitting that multiplication is the primary operation in the theory, the 
prevalence of exponentials in the formulae which we have developed in- 
dicates the way in which the subsidiary operation of addition arises. We see 
that in general the additive quantities of physics must occur in exponentials. 
An elementary example is afforded by action, which is well known to be 
additive; in wave mechanics it is represented as the phase angle of an 
imaginary exponential. Other cases are not so simple. We speak of the 
density p in a unit volume, due to the probability distribution of a particle, 
without specifying whether it is a low probability of a high mass or a high 
probability of a low mass; this agrees with the principle that the probability 
p and the characteristic m of the system are not to be detached from one 
another. If we consider a second particle, the combination of probabilities 
pp' involves taking the product pmp'm' =pp'. But the combined system is 
ordinarily considered to be characterised by a density p+p'* This is a point 
to be investigated in due course.! 

In so far as an addition a + j3 arises from a multiplication e" . eP> addition 
must be looked upon as a possibly non-commutative operation. We have to 

f The methods by which wave mechanics is adapted to treat additive properties instead 
of multiplicative properties are explained in Chapter xvr. 



1O2] Double Wave Vectors 155 

treat " ordered sums " of matrices as we treat ordered products. An integral 
may also require to be ordered. I do not propose to use the algebra of non- 
commutative addition, which is even less familiar than the non-commutative 
multiplication which we have employed; I mention it in order to show the 
type of mathematical complication in which we should be involved if we 
attempted to follow the common outlook more closely. 

Our first task is to study the double wave tensors which are formed when 
we multiply the simple wave tensors specifying the probabilities and con- 
figurations of two independent systems. 

10*2. Double Frames . 

By multiplying two wave vectors a , <f>p belonging to two independent 
systems we obtain a 16-valued quantity *F aj8 which we call a double wave 
vector. Formally it resembles a wave tensor of the second rank (mixed or 
covariant) obtained by multiplying wave vectors ^, Xp belonging to the 
same system; but it has a wider field of transformation. In a wave tensor a 
transformation of x * s locked to the transformation of ^r; but in a double 
wave vector there is no such restriction. If T is a covariant wave tensor and 
*F a double wave vector, their most general transformations are 



the former involving 16 coefficients g aj3 , and the latter 256 coefficients 



For example, let the objects described by the wave vectors 0, ^ be the 
earth and moon. We may use different axes of reference for the momenta, 
spins, etc., of the earth and moon. If we change the axes, there is no need to 
apply the same transformation to those used for the earth and those used 
for the moon. It is true that we do not usually avail ourselves of this liberty, 
and a transformation of axes would ordinarily apply to both bodies; but 
that is because we contemplate an earth-moon system, which is not 
merely a mental association of the earth and moon, but comprises physical 
relations between the two bodies not to be found in either body separately. 

More usually we are interested in the other aspect of a relativity trans- 
formation, in which it is regarded as a displacement without intrinsic change 
of a physical system, the frame being kept fixed. In the case of the earth and 
moon, this implies the introduction of two sets of vectors ^ a , fy capable of 
being rotated independently. 

To provide for independent transformations of ^ and ^ we must introduce 
two symbolic frames. We have therefore two complete orthogonal sets 
Ep, Fp. Inthematrixrepresentationof the symbols we take (forconvenience) 
Ep and F^ to correspond to the same matrix; but in chain multiplication 
EH is connected with the first suffix of a preceding or following double vector, 



156 Wave-tensor Calculus [10-2 

and Fp with the second. If more than one EorF symbol appears in a product, 
the jE's are in chain with the JE 9 s and the F'a with the F's. The order of 
writing E'B and F'B is of no importance; that is to say, every E^ commutes 
with every F^ . As usual E^ = i, -F 16 = i. 

Unless the contrary is stated we shall suppose that E^ and F^ are right- 
handed sets. It may sometimes be more appropriate to use a right-handed 
and a left-handed set; in this case the matrices E p and F^ cannot be identical, 
JF 16 and one of the pentads being reversed in sign as compared with the corre- 
sponding JS?-inatrices. The corresponding modification of the various 
formulae is easily found. 

Since (for right-handed sets) the letters E, F merely indicate the chain 
connection, there is no need to distinguish them when row-and-column 
suffixes are inserted. The E F notation is a device for extending the matrix 
notation, which omits suffixes, to expressions of the fourth rank such as 



The 256 double symbols EpF v (^, v= 1, 2, ... 16) constitute a complete set; 
that is to say, if a linear function of these symbols with algebraic coefficients 
is called an 2?.F-number, the operations of addition, subtraction and multi- 
plication applied to JfjP-numbers always yield ^J^-numbers. In matrix 
representation every E J'-number is a fourfold matrix of the fourth rank, 
and conversely. A matrix of the fourth rank T^ps is resolved into com- 
ponents according to the formula 



(10-221) 

or, without suffixes, T = S t E F v . (10-222) 

/*" 

The definition of pure wave tensors is extended to double wave tensors. 
A pure double tensor J is the product of two double wave vectors T, X*, 
and we have j. TO -E^ Vr . (10-223) 

A convention as to the order of suffixes in T or J is necessary. It is defined 
by (10-221). We write the product of the double vectors x F ajff , X y g* as / ay ,0s . 
The comma separates "first" suffixes (in chain with E^) from "second" 
suffixes in chain with F^ . 

To determine j^ v9 multiply both sides of (10-223) by E a F r and take the 
spur. The spur is formed by identifying the last suffix of each chain with the 
first suffix; thus the two chains contract separately. In particular we have 
spur (EpFJ^spm J^xspur F v . Hence, by (3-32), 



spur E u F v = if a or v ^ 16) 

M . \ (10-231) 

= ~16if/Lt = v=16. J v ' 

Hence spur p^J^F^FJ = spur (j^E^F^) - !#. (10-232) 



10-2] Double Wave Vectors 157 

By (10-223) this can be written 



Thus the formula corresponding to (3*37) is 

j^&X'E.F^. (10-24) 

Let the symbols E^F V in some conventional order be denoted by K a 
(or = 1, 2, ... 266). Then K a 2 = E^F* = 1, and K , K r either commute or anti- 
commute according to a fixed scheme. There are also multiplicative relations 
of the form K a K r = K^ or iK^ . These properties constitute the structure of the 
symbolic frame K^ , and any other set of symbols with the same structure is 
an equivalent frame. Equivalent frames are obtained by the transformation 



where q is any non-singular X-number. For, as shown in 2-7, this trans- 
formation does not change multiplicative relations. 

The theory of double frames K^, double wave tensors jF ay 0$, double 
strain vectors 5 ax ,0g, etc., follows on similar lines to those of simple frames, 
tensors and vectors. Since q now contains 256 components, we have 256 
independent relativity rotations, i.e. transformations without intrinsic 
change. The double space vectors are outer products of two complete space 
vectors, or sums of such products. We may call them complete space tensors 
of the second rank; but besides containing the ordinary components of a 
symmetrical tensor of the second rank, they include components of tensors 
of the third or fourth rank with some degree of antisymmetry (just as a 
complete space vector includes an antisymmetrical tensor of the second 
rank). 

We notice that the new matrices K^ given by (10-25) are not in general 
simple products of JF-matrices and .F-matrices. For example, if q = **!* 9 9 



This is evidently necessary because the K transformation has 256 coeffi- 
cients, whereas separate transformations of E^ and F^ are defined by 32 
coefficients. 

Considering a system described by a double wave tensor T, the trans- 
formation T f = qTq~ l , where q is a A-number, will represent displacement 
without intrinsic change. For, as in 2-9, the new system could have been 
obtained by carrying out the same construction in a different but equiva- 
lent frame K'qKq-*. This extension of the relativity principle is rather 



158 Wave-tensor Calculus [10-2 

difficult to grasp, because at first sight it seems to conflict with our ex- 
perience. If we rotate the moon without rotating the earth, we create an 
observable difference in the angle between their axes, which must surely be 
counted as an intrinsic change of the double system. The earth and moon is 
scarcely a fair illustration; but it is tempting to think that there would be 
an analogous intrinsic change in a system of two elementary particles 
(protons or electrons), since these possess planes of spin. Now it is clear 
that, if there is anything in the double system which is intrinsically changed 
by the transformation, it is unrepresented in the symbolic tensor T.f It 
signifies therefore that the conceptual process of combining systems is 
something other than a pure multiplication; in other words it introduces an 
interaction. We have already had a hint as to the nature of this modification, 
namely that in a combined system we contemplate only "simultaneous" 
configurations of the particles. In this chapter we shall not consider inter- 
action. Our point of view may be expressed by saying that we shall treat 
double systems as a preliminary to introducing the cementing interaction 
which will make them into combined systems. 

It has to be remembered that particles described by simple stream vectors 
have an even probability distribution throughout spherical space. Relations 
between them are therefore not very comparable with the relations between 
localised objects such as the earth and moon. It is one of the features of 
interaction that it renders possible a more localised probability distribution, 
as is illustrated in the theory of the states of a hydrogen atom. 

10*3. The Interchange Operator. 

Since the complete sets E^, F^ commute, they come under Case (6) of 2-7. 
They are therefore connected by a transformation 

, P 2 =l, (10-31) 



16 

and P = JSjE7 M ^. (10-32) 

i 



Since P 2 =l, PE^P^PE^P.PF^^F^. (10-33) 

And quite generally the operation P(...)P interchanges E and F within 
the bracket. 

Let y^-t(Vr + *r'|.). ^ = 4(V,-^). (10-34) 

Since P (E^F V E 9 FJ P = E^ E^F V 

on multiplying by final P, we have 



so that PY^=Y^P> PC |W --{ W P. (10-35) 



t It may be expressed in the matrix representation of T 9 since matrices have the property 
of symmetry or antisymmetry which is not invariant for transformations. We have seen 
( 7*3) that this discrimination corresponds to laying down planes of simultaneity. 



10-3] Double Wave Vectors 159 

We can write, instead of (10-222), 

2 7 =r-f2 7a = S^/y /iV + S/ MV ^ v , (10-361) 

where V^ + 'rfi* V"^ ""*** (10-362) 

Then PT=T*P, PT a =-TP. (10-363) 

That is to say T is divided into symmetrical and antisymmetrical parts 
T 8 , T a , which respectively commute and anticommute with the interchange 
operator P. 

There are 136 independent matrices y^ in the symmetrical part, and 120 
independent matrices ^ in the antisymmetrical part. 

Setting as usual T a j3=T^ a , we can show that 

PT=-f, Y*P=~T*. (10-37) 

By (10-31), PS^PV^F^V. That is to say, instead of applying tho matrix 
operator F^ to the second suffix of T, we can first apply the change repre- 
sented by P, apply the matrix operator to the first suffix, and then undo the 
change P (since P~ 1 = P). Evidently the operation P interchanges the two 
suffixes. It may include in addition a self-reciprocal operation Q which 
commutes with E^, so that QEpQ Ep. But since we may substitute any 
other J57-symbol or jP-symbol for E^ in the foregoing argument, Q must 
commute with all the symbols and therefore be algebraic. Thus Q= 1, 
since these are the only self-reciprocal algebraic operators. 

To determine the sign, consider the special case Y^ = 8 a , where S^ is the 
substitution operator. Then by (10-32) 



Using four-point matrices, the ten symmetrical matrices give 

V^=V=-^ 
and the six antisymmetrical matrices give E^E^ = 1. Hence 

SJ^^= -10 + 6= -4; 
so that P8=-l=:-8. 

Accordingly the sign in (10-37) is verified. 
The operator P can be factorised. Let 



We easily find 

i 

Alan P* 1 P21 PP PP 

xxIBO Xj *, -*2 "~" ' 12""" 2 1* 



160 Wave-tensor Calculus [10-3 

This factorisation is based on conjugate triads ( 3-8), and the factors 
P! , JP 2 are the interchange operators for the two minor complete sets. The 
operator P l is commonly written! 

A= -i(<W + <W + *s*s' + l)s (10-385) 

a form obtained by writing cr^ = t M in (3-87). P l is then treated as the whole 
interchange operator, the factor P 2 being replaced by its eigenvalue 1. 
This is legitimate in the more elementary types of problem for which a single 
set of Pauli matrices suffices. 

We can extend the theory of interchange to double sets. Let K^ L^ 
(JLI= 1, 2, ... 256) be two double frames arranged in corresponding order, so 
that Lp, L v commute or anticommute according as K^, K v commute or 
anticommute. By the method of 2-7, 2-8, we can find an interchange 
operator P KL , such that L^Pj^K^P^ 1 . ^ ^ e ^ are new symbols 
commuting with the JC M , the operator is 



256 

I % Vv < 10 ' 39 ) 



If, on the other hand, the L^ are If-numbers, the factor ^ is replaced by an 
adjustable algebraic coefficient a; also the singular case may arise, requiring 
us to substitute another reflection of K^ . 

10-4. Duality. 

Consider a matrix of the fourth rank which is the product of four wave 

V6Ct0rst ^-^Xr"*. (10-411) 

Since a and y are "first " suffixes, ^ and x are wave vectors in the frame E ; 
^ and o> are wave vectors in the frame F^ . Let 

x F = 0o>*, X = fo*, C7 = ^x*, F = ^o>*. (10-412) 

Then U and V are tensors of the second rank in the frames E and F 
respectively; X F and X, being composed of one vector from each frame, are 
double wave vectors. We have 



(10-413) 

We can associate *F and X with new frames Cj/ and D^' in the same way 
that U and V are associated with the frames E^ and F^ . When the "crossed 
frames" G^ and D^ are adopted, Y and X become wave tensors of the 
second rank and 17 and V become double wave vectors. 

This gives two alternative resolutions of T into matrix components, viz. 
T = S^J^P V =S V / C' /1 / JD; (10-421) 

t Dirac, Quantum Mechanics, 2nd ed., p. 226, equation (35). Dirac uses Pauli matrices 
whose square is + 1. 

| It is not necessary that the factorisation should be actually possible; the theory in this 
section will apply to symbolic factors ( 3-3). 

Unaccented frames C 9 D are introduced later. 



10-4] Doubk Wave Vectors 161 

or, with suffixes, 

2^8 = S fa (E^ Y (E w )p - S V ' (Z^U < jy fr . (10-422) 
The distinctive notation C", D', E, F is not required when suffixes are 
inserted. 

Evidently a double matrix E^F V is a linear function of the 256 symbols 
Cp'ty, and vice versa. We shall determine this function explicitly. Let 



Or, With SUffixeS, (^) ay (JFJ / M = S a-T V W (^rU(^r)jiy- ( 10 ' 432 ) 

Multiply both sides by (^\)8 ( E P )yp The right-hand side is 



The left-hand side is 

(^W 
Hence 

^A P = TVspr(^^^S A ). (10-44) 

As an important example, consider J5 16 F w . Let 



or, with suffixes, -(l) ay (l)^ = S ff . T ^ar(^)8(^r)fr. (10-452) 

Then, by (10-44), '= -&spur(E T E ff ) (10-46) 

o that v' aa = J, v'er = (cj ^ T). Hence 



where P OD ' is the interchange operator of the frames C^' and /}/. Similarly 

OJD^-Pn. (10-472) 

Thus by crossing the frames, algebraic quantities in the frame C'D' 
become multiples of the interchange operator of the frame EF, and vice 
versa. 

We find similarly that E tL F fl = Z a (C a 'D a '), (10-48) 

where the minus sign is to be taken if E a anticommutes with E^ . 

A slight variation of the foregoing process is obtained by taking frames 
Op , Dp such that the connection 

2 7 = S^^ V = S V C^Z) V (10-491) 

stands for T a p - S fa (E^ (E v )& S v (^) aj3 (^) y8 . (10-492) 
This "straight cross" is easier to manipulate analytically, and we shall 
generally use it; though it is perhaps less physically significant than the 
"inverted croBs". If j^=S V>OT <7 Z> T 

we obtain, by the same method as before, 

J r^r). (10-493) 



162 Wave-tensor Calculus [10-4 

where, as usual, (jE T ) aj8 =(JS T )^ a . Hence 



the minus sign applying to the six antisymmetrical (time-like) matrices. 

Ah V 4 5 - IS ( CM ( 10 ' 495 ) 

the minus sign applying to the six electrical matrices a = 01, 02, 03, 04, 05, 16. 
The same formulae hold if EF and CD are interchanged. 

We call the tensor r^ the dual of the tensor t^ v . 

Denoting the double matrices E^ F V9 C^D V by K a , L a (<r= 1, 2, ... 256), the 
equivalent frames K a and L a are connected by a transformation 

L a =qK a q~i. (10-496) 

It should not be difficult to evaluate q as an E jF-number or CD-number; but 
I have not succeeded in doing so. The calculation is complicated because it 
is found to come under the singular case; the value of P KL calculated from 
(10-39) is zero, so that q^P KL - *&*& same applies to the transformation 
connecting E tL F v and C^DJ. 

Without actually evaluating q, we see that the crossing of frames is 
included in the relativity transformations of the double frame. We can 
regard the transformation as a continuous one, giving a sequence of double 
frames intermediate between EF and CD. 

10*5. Significance of the Grossed Frames . 

According to the uncertainty principle there are two extreme ways of 
specifying a particle: (a) the momentum factor may be specified exactly, 
so that the position vector is entirely uncertain, and (b) the position vector 
may be specified exactly, so that the momentum vector is entirely uncertain. 
Wo shall find that the effect of crossing frames is to transform specification 
(a) into specification (b). That is to say, a particle which has the specification 
(a) in the EF frame has the specification (b) in the CD frame, and vice versa. 
Double frames enable us to express symmetrical space tensors of the 
second rank in wave tensor form. (A single frame provides only for the anti- 
symmetrical tensors of the second rank.) Consider, for example, the outer 
square p^p,, of a momentum vector jp M . We call Ppp the energy tensor, f 
or more precisely the self -energy tensor of the particle. In ordinary relativity 
theory the energy tensor of a particle is j^Pv/w; but the factor m belongs to 
a later stage in the adaptation of our equations to practical conditions, and 
can be omitted for the present. By (6-51) the four-dimensional momentum 
vector is represented by EE^p^ and the energy tensor is therefore repre- 
sented by ZXffrtpp, Oi, v = 1, 2, 3, 4). (10-51) 



t Both the momentum vector and the energy tensor include momentum and energy. It 
is an accident of current nomenclature that they are named differently. 



10-5] Double Wave Vectors 163 

If the momentum vector is exact, we may take its direction as our time 
axis. The energy tensor (10-51) then reduces to a single component 

aJS? 45*45- (10-521) 

To represent an entirely uncertain momentum we must distribute the pro- 
bability evenly over momentum vectors in all directions in space-time. The 
resultant momentum will be zero, but the resultant energy tensor will be 
definite. "Entire uncertainty" implies that there is no preference for any 
direction in space-time; the resultant energy tensor must therefore be of a 
form invariant for rotations and Lorentz transformations. The only in- 
variant tensor of the second rank is S^. Hence 

s^^-jsy, 

where j8 is an invariant, and the summation extends over all the vectors 
which compose the probability distribution. Then by (10-51) the resultant 
energy tensor is fi ^ ., 

P ( A i6*i5 + #25*25 + ^35*35 + ^45*45)- (10-522) 

To extend this from simple space vectors to complete space vectors, we 
define the complete self -energy tensor to be the outer square of the complete 
stream vector. If the complete stream vector is exact, it can be reduced 
to the form a* (E l + E 2 ^+ 46 + E u ), and the complete energy tensor is 

a (^ + EM + # 46 + E u ) (1\ + JF M + F^ + JP 16 ). (10-523) 

This refers to an elementary charged particle; for a neutral particle the 
complete tensor is still of the form (10-521). 

If the stream vector is entirely uncertain, the distribution must be such 
that the resultant energy tensor is invariant for all appropriate relativity 
rotations. The natural generalisation of (10-522) in 

10 
02^*;. (10-524) 

But this corresponds to a probability distribution which is symmetrical 
with respect to an electrically saturated space-time. To obtain symmetry 
with respect to neutral space-time, we must substitute neutral sets (6-412). 
Distinguishing kinematical and electrical matrices by suffixes k and e, the 
resultant complete energy tensor is then 

^(S A ^^~S e ^^) = 4j8(7 45 jD 45 (10-525) 

by (10-495). Comparing with (10-521) we see that this corresponds to a 
neutral particlef with definite stream vector in the CD fiame. 

This establishes the important result that for a neutral particle an entirely 
uncertain distribution of momentum in the EF frame corresponds to an 
exact momentum in the CD frame. (The momentum is here understood 
to include energy, spin and magnetic moment.) 

t When the complete uncertainty extends to all components of the stream vector, the 
charge must be completely uncertain. 



164 Wave-tensor Calculus [10-5 

We see also that the straight cross CD is involved because we take a 
neutral frame as standard. The theoretically simpler inverted cross C'D' 
corresponds to a saturated EF frame. 

The position vector can be treated similarly. We can define a complete 
position vector which includes, for example, the angular coordinates con- 
jugate to the spin momenta. For a neutral particle it reduces to the ordinary 
four-dimensional position vector, since there is no spin or magnetic moment 
to be described. The outer square of the position vector will be called the 
position tensor; it plays an analogous part to the energy tensor in the fore- 
going theory. 

The energy tensor and the position tensor have definite values even 
when the corresponding vectors are uncertain; but for brevity we shall 
call those of type (10-521) "exact", and those of type (10-525) "uncertain". 
If a particle is represented by an exact energy tensor and an uncertain 
position tensor in the EF frame, it is represented by an uncertain energy 
tensor and an exact position tensor in the CD frame. If the scale relation 
is appropriately chosen, the exact energy tensor of the EF frame is identical 
with the exact position tensor of the CD frame; and similarly the two un- 
certain tensors are identical. In short the position space of the EF frame 
is the momentum space of the CD frame and vice versa. 

We have here adopted the four-dimensional point of view, in which the 
position vector is measured from an origin in space-time. This was necessary 
in order to compare our formulae with the ordinary form of the uncertainty 
principle; but it slurs over the effect of curvature of space-time.t In five- 
dimensional representation the position vector is measured from an origin 
at the centre of curvature of space-time; and in the standard notation an 
exact position tensor has the form 

a^ 5 ^ 6 = JaS(C' /A Z{ i ), (10-53) 

where the minus sign refers to p = 04, 14, 24, 34, 54, 16, by applying (10-493). 
The connection between the position space and the momentum space is, as 
it were, rotated by the matrix JSJ 4 . The connection is expressed more simply 
by correlating the four-dimensional density of the position vector to the 
three-dimensional density of the momentum vector. Each of these gives a 
second-rank tensor of the form J 

*tfi6*i0= i (S.C^- S^Zp, (10-64) 

t If the position were entirely uncertain in infinite space, the position tensor would be 
infinite. In assuming it to be finite, we have in a sense taken account of curvature of space- 
time; but it has not been necessary to introduce the curvature in any other way. 

% Multiplying the position vector (matrix E 6 ) by iE 6 to obtain its four-dimensional 
density, and multiplying the momentum vector (matrix E t6 ) by iE^ to obtain its three- 
dimensional density, the result is in each case an algebraic quantity (matrix E u ); so that 
the corresponding second rank tensor is &E U F 16 . 



10-6] Double Wave Vectors 165 

where s and t refer to the space-like and time-like matrices. It is these 
densities which become interchanged in the transformation EF-> CD. 

Formally an energy tensor ^ t E tL F v p VL p v refers to two particles or systems, 
since it introduces two frames. It is a special case of a mutual energy tensor 
of two particles ^E^Fyp^p^ formed by multiplication of their stream vectors 
SJS^pp, I,F v p v ', as the combination of probabilities requires. A self -energy 
tensor is obtained by imposing the constraint p^ 'p^' (or alternatively 
PH~ JP//); this constraint greatly limits the number of relativity trans- 
formations applicable. There are two possible interpretations of such a 
constraint; either the second particle is a fictitious duplication of the first, 
introduced for formal purposes, or the centroid of the two particles is 
deemed to be fixed so that their stream vectors are automatically made 
equal and opposite. 

This study of a double frame throws considerable light on the relation of 
position and momentum in the matrix theory. But it is only a step towards 
the actual conditions in which position and momentum are known obser- 
vationally; and further progress will be made in Chapter xi. 

10-6. The 136 -dimensional Phase Space. 

When four-point matrices are used, the frames JE^, F^ each contain ten 
imaginary and six real matrices. Hence the frame E^F V contains 

(10 x 10) + (6 x 6) = 136 real matrices 
and (6 x 10) -f (10 x 6) = 120 imaginary matrices. 

Since (E fl F v ) 2 =l 9 the eigenvalues of E^F V are real. Accordingly, by the 
definition in 7-3, the 136 real matrices are space-like and the 120 imaginary 
matrices are time-like. 

The general infinitesimal transformations for covariant final and contra- 
variant initial double wave vectors are 

T'-e^T, X*' = X*e-*, (10-61) 

where d&^^K^O^. Considering a single term 0^, the two transforma- 
tions in (10-61) are the same if 

exp&(^)r(E v )M.V^^ 
That is, if - ( jy ya (E 9 )y - (E^ y (E v )p 8 

or -KpF^EpF,. (10-62) 

This requires that E^F V should be imaginary or time-like. Thus when E^ F v 
is time-like, a covariant wave vector transforms like a contravariant wave 
vector, and a covariant wave tensor transforms like a mixed wave tensor. 

Proceeding as in 7-2 we obtain double strain vectors (covariant double 
wave tensors), which behave like double space vectors for time-like trans- 
formations and substitute antiperpendicular rotations for relativity 



166 Wave-tensor Calculus [10-6 

rotations in space-like transformations. The latter generate a 136-dimen- 
sional phase space. 

The theory of ten-dimensional phase space applies without important 
modification to 136 dimensions. Stereographic coordinates can be intro- 
duced; and the result corresponding to (7-59) is 

V^7= .R- 136 (1 + r 2 /4JR 2 )- 136 . (10-63) 

Since (EpF v ) 2 =l, the circular rotations are of the form e iE ^ fvU ^, where 
Up V is real. Thus the matrix d 8 for real displacements in phase space is 
imaginary, as in simple phase space (7-31); but the coefficients 0^ (unlike 
the 0^) are imaginary. 

The condition E^ F v = E^F V satisfied by the space-like matrices means that 
they are symmetrical for an interchange of the suffixes a, )8 with y, 8 in 
(10-492). There is another kind of symmetry which corresponds to the inter- 
change of a, y with j8, 8, i.e. interchange of first suffixes with second suffixes. 
The double matrices y^ v introduced in 10-3 have this kind of symmetry, 
and the ^ have the corresponding antisymmetry. It is inconvenient to 
work with y^ v and J^, since they have no simple commutative properties 
and their squares are not algebraic. But the two kinds of symmetry are 
interchanged by crossing the frames as in (10-492); hence the space-like and 
time-like matrices of the CD frame have the same symmetry and anti- 
symmetry as the y^ v and l^ v . It is easily seen that the 136 y^ v are linear 
functions of the 136 space-like matrices of the CD frame and the 120 
are linear functions of the 120 time-like matrices. 

Thus the separation of T into T 8 + T a in the EF frame (10-361) is the 
same as the separation into space-like and time-like parts in the CD frame. 
For any double system we have two alternative phase spaces according 
as we adopt the EF frame or the CD frame. If one of them corresponds to 
classification of configuration by position, the other will correspond to 
classification by momentum. If a simple E or F phase space gives a classi- 
fication by position, the EF phase space will give the position configurations, 
and the CD phase space will give the momentum configurations. For 
example, an algebraic displacement in EF space represents change of time, 
and in the CD space represents change of energy. Since the points of CD 
space represent distributions with fixed momentum, generally called 
elementary states, we may regard the CD space as composed of states and 
the EF space as composed of configurations. 

It is perhaps not self-evident that the EF space gives the same kind of 
classification as the simple E and F spaces. t For a crucial test consider 
an electrical displacement with matrix E l for the space vector or j 23 for the 
strain vector. To give this displacement to both particles we must apply 

t It had occurred to me that the process of amalgamating two simple systems to form a 
combined system might involve crossing their frames. The test here given dispels this idea. 



10-7] Double Wave Vectors 167 

the transformation q**eto*+**t 9 to the double strain vector. Since this is 
a y 2 3,ie rotation it gives displacement in CD space; there is no displacement 
in EF phase space (or in the separate E and F phase spaces) since the 
matrices E 2S9 F Z3 (or H n F M9 S l9 F n ) are time-like. It can scarcely be sup- 
posed that we create a positional displacement of the double system without 
displacing the simple systems positionally; but the displacement in CD 
space can well be interpreted as a change of electrical energy, which has no 
counterpart in a simple system. This indicates that the E F phase space in 
which no change occurs is the positional space, and the CD phase space is 
the momentum space. 

The following property of CD phase space is of interest. If we divide a 
double wave vector into symmetrical and antisymmetrical parts 1 F*, X F, so 
that X F = Y*, Y= - Y, we have, by (10-37), 

H* = i(l-P)Y, Y*=H1 + P)Y. (10-64) 

Consider a transformation V = <f. If q commutes with P, we obtain 

( x F')* = g x F, (')=}. (10-65) 

But for a non-commuting g, say q = 1 + ^,,0, 



(10-60) 

Thus *F* and ^ a are kept distinct in the y^ v transformations but not in 
the Jp,, transformations. The former comprise all the displacements in CD 
phase space. Thus CD phase space has the distinctive property that a strain 
vector of the form Y*O* remains of the same form at all points of phase 
space; similarly for the forms x F"O a , *p<I> a . Thus states determined by 
symmetrical and antisymmetrical wave vectors X F*, *F a form distinct sys- 
tems not transformable into one another by strain. 

This separation of symmetrical and antisymmetrical wave functions plays 
a prominent part in the current treatment of double systems, as developed 
by Fermi and Dirac. Since it applies to CD space but not to EF space, it 
applies to states not to configurations. Since this is in accordance with the 
current interpretation, it confirms our identification of the two phase spaces. 

10*7. Representations of Two Charges . 

The strain vectors representing elementary charged particles have been 
given in (6-64). Using the matrix representation in (3-61), we resolve them 
into their wave vector factors 



8 C =m (1, - 1, -i, i) (1, - 1, i, -i), 

5jm(l, - 1, i, -i) (1, - 1, -*, i). (10-71) 



168 Wave-tensor Calculus [10-7 

First consider two charges of opposite sign and the same spin, whose 
wave tensors may be taken to be 



so that (omitting the masses m, m') we have 

= w = (l, 1, -i 9 -i), Xss j = (i 9 i f t,i). 

The double vectors, obtained by a straight cross, are therefore 

T = ^* = ^, X = X o>* = ^. (10-72) 

If the masses are inserted, T and X can be given any coefficients m", m'" such 
that ra"w'" = mw'. 

As another example take two charges of the same sign and spin, with 



Then Y = ^* = (l, 1, -i, -)(!, 1, -i, -i) 

=1 i -i -; 

1 1 -i -i 



so that the double wave vectors are degenerate (5- 56). 

A combination of two charges of like sign and opposite spin, or of opposite 
sign and spin, likewise gives degenerate double wave vectors. 

We see that there is a simplicity in the combination of charges of opposite 
sign with like spin, which is not exhibited by any other combination. The 
CD and the EF representations are interchangeable, and the double wave 
vectors are non-degenerate. The physical interpretation of this peculiarity 
is that when charges of opposite sign with like spin are superposed, their 
electric and magnetic fields cancel. They thus constitute a self-contained 
unit which can be inserted in a background of neutral space-time without 
disturbing its neutrality. Other combinations cannot be treated as isolated 
units for they disturb by their external fields the surrounding matter. In 
their case we must take into account, in calculating probabilities for the 
purposes of statistical theory, not only the probability of the configuration 
of the two particles, but the probability of the polarised configuration of the 
rest of the universe. 

If then we wish to treat what is strictly a two-body problem, we must 
confine ourselves to particles of opposite sign and like spin. 

To understand the nature of the superposition referred to above, we must 
recall that the momentum vectors of the two charges are given definitely 



10-75] Double Wave Vectors 169 

by S a and S b . Hence, by the uncertainty principle, their positions are 
entirely indeterminate. We are therefore superposing two probability 
distributions which extend uniformly over the universe. It is not a question 
of superposing localised particles; they would be disturbed by interaction 
effects which we have not yet studied; and their exact superposition is, in 
fact, inhibited. The theory of this chapter is limited to the combination of 
independent probabilities. 

It will be seen from (10-71), or directly from (tf-64), that 



By making an inverted cross we obtain double vectors l l'" = 0w*, 
X' = <x*. These are non-degenerate if the two charges are of like sign 
and spin, and are degenerate for all other combinations. We have already 
noticed that the inverted cross bears the same relation to electrically 
saturated space that the straight cross bears to neutral space; and the 
non-degenerate combination of two charges of like sign and spin would 
evidently fit saturated space in the same way that two charges of opposite 
sign and like spin fit neutral space. 

10-75. Importance of Double Tensors. 

The fundamental tensors of relativity theory are symmetrical space tensors 
of the second rank, viz. the metrical tensor g^ v and the energy tensor 
G^- Ig^O. These must appear as double wave tensors in the wave-tensor 
calculus; and it is clear that any serious attempt to unify relativity theory 
with wave mechanics must be based on double wave tensors. In macro- 
scopic relativity theory the interval ds between two points is regarded as 
the fundamental observable; correspondingly in relativistic wave mechanics 
the double wave tensor of two particles is the basis of description. 

In treating a system of n particles, we naturally introduce a corresponding 
n-tuple wave tensor. But the step from a double to an ra-tuple wave tensor 
is not comparable with the step from a single to a double wave tensor. In 
relativity theory we do not recognise any relation between three points 
analogous to the interval between two points; nor are there any space 
tensors of the nth rank analogous to the metrical tensor and the energy 
tensor. In proceeding from simple to double wave tensors we approach the 
meeting point with ordinary relativity theory; in proceeding to treble and 
n-tuple wave tensors we diverge again. Quantum physicists have them- 
selves recognised that the specification of a system of n particles cannot well 
be left in the form of an w-tuple wave tensor, and have introduced another 
method of specifying such systems by a sum (not product) of Jordan- Wigner 
wave functions, which will be used in Chapter xvi. 

The number 136, being the number of dimensions of double phase space, 
will be prominent in our formulae for the principal constants of nature. This 



170 Wave-tensor Calculus [10-75 

does not imply that their application has special reference to systems con- 
sisting of two particles. It implies that in wave mechanics, as in relativity 
theory, the study of a complex system is based on the relations between pairs 
of particles contained in it, and on the tensors of the second rank which 
embody these relations. The number of dimensions of double phase space 
occurs in the formula for the mass of an electron or proton, because mass is 
by definition contained in a space tensor of the second rank equivalent to 
a double wave tensor. 

The quadruple tensor also plays a rather special part in the theory. If a 
primitive entity or relatum is represented by a simple wave tensor, we require 
two relata in order to provide an observable relation. To assign measure to 
this relation we require another observable relation with which to compare 
it. Thus a measure involves four relata two to provide a quantity to be 
measured and two to provide a unit of comparison. It therefore depends on 
a quadruple wave tensor. This coincides with our conclusion in general 
relativity theory :f 

Thus four points is the minimum number for which an assertion of absolute 
structural relation can be made. The ultimate elements of structure are thus 
four-point elements. 

Thus simple, double and quadruple wave tensors appear in the theory; 
but there is no occasion for introducing any other combination for the 
purposes of fundamental study. For systems of more than four particles 
(or two particles plus a "comparison fluid") we abandon the method of 
multiplication of probabilities, and develop a procedure (formally investi- 
gated in Chapter xvi) which is in a sense intermediate between macroscopic 
theory and elementary wave mechanics. 

Generally the quadruple wave tensor remains in the background of the 
theory; we need only occasionally remind ourselves that the double wave 
tensors, which we employ, require such a background. We shall generally 
regard the quantities described by double wave tensors as observables, in the 
same way that the interval ds is usually regarded as an observable although 
any numerical value that is attached to it expresses its relation to another 
interval. 

10-8. Relative Coordinates . 

Let Xp, Xp be the coordinates of two points in four dimensions, associated 
respectively with symbolic frames E^, F^ . We make the transformation 



which gives x^ = A^ + f^ , x^ = X M - ^ . (10-812) 

t Mathematical Theory of Relativity, 98. 



10-8] Double Wave Vectors 171 

Then the x^ are coordinates of the centroid, and 2 M are the coordinates 
of one point relative to the other. The corresponding formulae for the 
derivatives are a a 9 a 3 a 

a^-gi' + aV' K^T'top top" 



fi x nl^_. ' i + It -3 /~nl3,- 3 f (lU-0 w) 

We are going to find two new complete sets G^ , H^ which, we shall show, 
are associated with x^, ^ in the same way that E^, F^ are associated with 
x n> x u> so ^at IL is the frame whose rotations determine the vector 
transformations of the relative coordinates 2^ . 

Let P be the interchange operator ( 10-3) of E^ and J5J, , and let 



for p,= 1, 2, 3, 4. We here write V for an algebraic square root of 1, not 
necessarily the same square root as E 1B and J^ 16 . Solving (10-831) for E^ 9 F^ 
we find p p 

^rS^-*" 7 ^' ^-fr?^-^- (10>832) 

Then 



Similarly 0^0, = (1/2*0 

(10-841) 



Since ^, v are restricted to the values 1, 2, 3, 4, the right-hand side is anti- 
symmetrical in p. and v. Therefore G^ G v =-G v G fA . Similarly H^ H v = - H v H^ . 
Also 



(10-842) 
These two expressions are equal, so that O^H^ I^G^. 

These results show that G l9 2J G 39 C? 4 and 15^ , // 2 9 H.^H^ are tetrads, and 
that the H'B commute with the (?'s. From these tetrads we construct two 
complete sets 0^, jfi^ (fi= 1, 2, ... 16) which commute with one another. By 
(10-842) GHl2i f )(F. + i f E A )^E l F A (10-843) 



for p, = 1, 2, 3, 4. This result can be extended to all values of ^; e.g. 
G^H^G^^H^G^.G^^E^.E.F^E^F^. 

Hence by (10-32) the interchange operator P for E^ and J^ is also the inter- 
change operator for G^ and H^ . 



172 Wave-tensor Calculus [10-8 

The Gp and H^ not belonging to the original tetrad do not obey (10-831). 
In general they have no simple connection with the corresponding E^ and 
F^\ but the result for G b , # 5 is of interest. We have 



\iP (Ei + VPJ P (E 2 + t' 



2 EM\-i'EiFi-EuFu-i'E u Fu-EsFs-...). (10-861) 

The bracket contains 16 terms. We have given one specimen of each type 
of term . All terms of the same type have the same coefficient ; the verification 
of this is an interesting study of the operation of the permutations. We 
obtain similarly from (10-8.5) 



(10-862) 
Multiplying (10-861) by E$ and (10-862) by i'F b and adding, we obtain 



Hence, multiplying initially by P and finally by G 5 , 

#5 = -(J^ + t'JU- -G f ', (10-863) 



where G 5 ' is the symbol defined analogously to O l9 G 2 , G 3 , 6? 4 by putting 
ft = 5 in (10-831). A similar result is obtained for H 5 . 

Thus we can, if we wish, extend the present investigation to relative 
coordinates in five dimensions, defining 6?^, ff /x (/x= 1, 2, 3, 4, 5) by (10-831); 
but in that case if E^, F^ are right-handed sets we must construct G^, H^ 
as left-handed sets. Since we have at present no use for this extension, we 
keep to right-handed sets, and accordingly define 6? 5 as i6r 1 6r 2 (7 3 6r 4 . 

By (10-811) and (10-831) we have 



4 4 

Or writing x = S E^ , x = S G^ x^ , etc., 



(10-871) 



(10-872) 

For example, if we have two non-interacting particles of proper mass m 
which satisfy the wave equation (8-262) without electromagnetic field, viz. 



Similarly, if V a = 2^3/9^, V i= = 2^9/3^, etc., 
i ^ i 

op 
' 



10-81 Doubk Wave Vectors 173 

the product wave function T satisfies both equations, and therefore satisfies 



Hence, multiplying by 2P/(1 + i'), 

0. (10-88) 



A noteworthy point is that the vectors x, x', and also the vectors x, 5, 
have to be combined with a quarter period difference of phase represented 
by i'. There is evidently no other way of obtaining a transformation of 
the kind required. If we try to combine them without phase difference, 
we have 



The trouble then arises that E^ + F^, E^-F^ are singular symbols with 
eigenvalues and 2i. The product of two vectors (E + -P\) x^ , (E - 1\) & 
in the same direction in the absolute and relative spaces is identically zero. 
This means that we have made a singular transformation, so that either the 
external (absolute) configurations or the internal (relative) configurations 
have shrunk to a point in our representation. 

In the transformation here found A-^ and ^ are referred to symbolic 
frames which are equivalent to one another and to the original frames of 
x and x'. Thus the relative coordinates can be treated in the same way as 
the absolute coordinates hitherto treated. It would be more logical to say 
that the absolute coordinates can be treated in the same way as relative 
coordinates, for all our experience is concerned with relative positions. But 
to render the representations similar we have had to measure the co- 
ordinates in various complex units. This is seen by writing (10-871) in the 
form - 



There does not seem to be any way of extending the foregoing transforma- 
tion to particles of unequal mass. On reflection we see that no such extension 
could be expected. The integrated mass of a particle distributed uniformly 
all over the universe is a highly artificial conception, and has no clone 
relation to the concentrated masses of classical particles. Particles with 
different masses (protons and electrons) do not appear in the theory until 
a later stage of development than that which we have now reached, and their 
wave functions are not the primitive relativistic wave functions, introduced 
by Dirac's equation, which we are at present studying.! 

In the next section we treat a more familiar transformation of a somewhat 
different character which concerns particles of unequal mass. 

t See 11-6. 



174 Wave-tensor Calculus [10-9 

10*9. Relative Coordinates Unequal Masses. 

For two particles of proper mass m, m', the coordinates x^ of the centroid 
and the relative coordinates f^ are given by 



Hence m 38 m' 3 ___ d_ no .Q 12 \ 

l ' 



Writing M=*m + m', p, = mm'/(m+m f ), (10-92) 

we obtain from (10-912) the well-known formula 

IJL+JL^I^.+AJL (10 -93) 

Introducing momenta, ^= id/dx^, m = idfi^, etc., this becomes 

In the transformation (10-911), the jacobian 

|i5f^=l (10-95) 

so that the transformation x^ , x^-^x^ , ^ does not alter volume elements. 
Normally (10-911) applies to space-coordinates only, the time being the 
same for the two particles. We can, if occasion arises, extend it formally to 
four coordinates; but it must then be remembered that the relative time 
r = t t' is not the time which is ordinarily associated with relative co- 
ordinates f ^ . 

We notice especially that 

mm' = Mp. = M , say. (10-96) 

We have seen ( 10-1) that a system is to be regarded as the product, rather 
than as the sum, of its parts; and from this point of view the product mass 
M is its leading dynamical characteristic. The two ways of factorising M 
lead to two modes of dividing the system either into two ordinary particles 
m, m', or into an external particle M and an internal particle p. These two 
modes of factorising M are familiar in classical mechanics, especially in 
celestial mechanics, where p is called the reduced mass associated with the 
relative orbit. 

If p 2=s Pi 2 +p2 2 +Pa*> etc., we obtain by summing (10-94) 

p 2 /m +p' 2 /m' = P*IM + t& 2 //*. ( 10-97) 

For simple plane waves the energy was found in (5-251) to be 

/n *?) /^/ //>* 2 _L /yi2\i _o_ <**i _L i/n2//*yi /lO*Qfil\ 

The conditions described by plane progressive waves are, however, highly 
abstract, and (10-981) has no immediate interpretation in terms of obser- 



10-9] Double Wave Vectors 175 

vation.f All observed motion is relative motion, and the energies with 
which we are concerned observationally correspond to relative coordinates 
1^. Clearly energy must be defined in such a way that either mode of 
division of the system gives the same total energy. By (10-97) the function 
of p, which is conserved in the transformation from P, w to p, p', is p 2 /ra; 
the variable part of the energy must be equal or proportional to this. We 
therefore set the energies equal to 



, 

where a is a constant coefficient. These satisfy 



(10-983) 

We notice in (10-982) that the relative energy e does not include any rest- 
energy. 

It has commonly been supposed that the energy m + op 2 /w is merely an 
approximation to (m 2 + jp 2 )*, the value of a being \ . That is not so ; m + ojj*/m 
is the exact expression in the conditions with which we are here concerned, 
and (10-981) is irrelevant. According to the common assumption tho 
variable part of the energy is {(m 2 +p 2 )l m}, and it is taken for granted 
that the relative energy is also of this form, viz. {(p 2 + iD 2 )^ //,}. The latter 
expression is the one which concerns us observationally, since we can only 
observe relative positions and energies. But 

{(m 2 +# 2 )* - m} + {(m /2 +y ^ 

Thus the sum of the internal and external energies of the system is not 
equal to the sum of the energies of the constituent particles ; and it is 
clear that the form used is inadmissible. 

We shall have more to say on this point in Chapter xni. Meanwhile the 
following explanation must suffice. The expression for the energy operator 
in terms of the momentum operators p l9 p 2 , p 3 (the three-dimensional 
hamiltonian) is a leading dynamical characteristic of a system. What we 
here find is that the two parts into which we divide a system have not the 
same dynamical characteristic as the set of plane waves studied in 5-2. 
It was scarcely likely that they would; for our first investigations were 
concerned with abstract conditions, and it is only with the introduction of 
double wave functions that we begin to make some approach to actuality. 
In order that the wave equation #0 = of a simple wave vector may be 
invariant for rotations of space-time, H must be a space vector. Corre- 
spondingly, if the wave equation of a double wave vector is // 2 *F = 0, // 2 

f Its application to the observed change of mass with velocity of an electron cannot be 
described as immediate, since the mass is distributed over an infinite wave front. To connect 
this with the change of mass of an approximately classical particle it is necessary to in- 
vestigate the transformation of the normalising conditions. 



176 Wave-tensor Calculus [10-9 

must be a space tensor of the second rank. If we divide H 2 into two parts 
H 2 ' + H referring to the two particles, H 2 ' and H% may or may not be 
tensors separately; but at any rate they will not be space vectors. Thus the 
vector form of H, which yields p Q = (m a -f jp 2 )*, is not applicable to the con- 
stituents of a double system; and p*/m occurs in (10- 982) in its own right as 
a component of a tensor of the second rank, not as an approximation to a 
component of a vector. 



PAET II 
PHYSICAL APPLICATIONS 



CHAPTER XI 
THE RIEMANN-CHRISTOFFEL TENSOR 

11-1. The Comparison Fluid. 

When the position of a body is determined by observation it is referred to 
landmarks furnished by material objects; but in theoretical formulae the 
position is supposed to be specified relatively to a geometrical frame. In 
macroscopic physics the geometrical frame is an admissible substitute for 
material reference marks; and there is no inconsistency in assuming that 
the position, momentum, etc., defined by reference to points of a geometrical 
frame, satisfy the same equations as the position, momentum, etc., measured 
from material landmarks. But it is not so in wave mechanics. A material 
frame of reference cannot be equivalent to a geometrical frame. For the 
material system is specified by a wave function which describes a pro- 
bability distribution of its observable landmarks relative to a geometrical 
frame. To identify the position and motion of the landmarks with the 
position and motion of a geometrical frame contradicts the uncertainty 
principle. 

Thus in so far as the equations of wave mechanics relate to anything 
observable, they contemplate (a) a geometrical frame, (6) a physical refer- 
ence object or system of objects, and (c) the particular particle or system 
under discussion. The geometrical frame cannot be omitted without aban- 
doning the whole method of wave mechanics, but it is only an intermediary 
and not the final reference system for our observations; (6) and (c) are neces- 
sary in order to furnish observable phenomena for measurement. For 
simplicity the casual reference objects (6) used in actual experiments are 
replaced by an ideal standardised reference object. The fundamental 
equations of physics necessarily contemplate highly idealised systems in 
highly idealised conditions; and they select a reference object with simple 
and symmetrical properties, just as they select for investigation very 
elementary systems (c). But the idealisation must not be carried so far as 
to substitute something which 'has not the same relation to our sensory 
experience. Idealisation is permissible; abstraction is not. The practice in 
current quantum theory of substituting a sharp geometrical frame for a 
probability distribution of observable landmarks is clearly indefensible. 

Probably it has been thought that the difference between (a) and (6) 
would be made insensible by using very massive material landmarks. But 
a massive system causes curvature of space. Current quantum theory 
neglects curvature of space, and therefore falls into error either way: either 
it postulates that a light reference object is used and wrongly neglects its 

X2-2 



180 Physical Applications [11-1 

uncertainty of position and velocity, or it postulates a heavy reference 
object and wrongly neglects the resulting curvature of space. Perhaps the 
most important insight, obtained through a combination of relativity 
theory and wave mechanics, is a realisation that the two alternatives are 
different forms of the same error. 

The magnitude of the error introduced by failing to distinguish (a) and (6) 
is a matter for detailed calculation, and will be found in due course. We need 
only say here that the common impression that it is likely to be trivial is 
very far from the truth. 

The idealised physical reference object, which is implied in current 
quantum theory, is a fluid permeating all space like an aether. Such an 
aether is in a sense a materialisation of space; or better, space (the geo- 
metrical background or metrical field) should be looked upon as a de- 
materialised abstraction of the adopted comparison aether. " Dematerial- 
isation" is represented analytically by replacing a probability distribution 
by a sharp configuration. The uncertainty principle does not allow such 
replacement until we have removed from the comparison fluid those 
characteristics which make it accessible to observation so that it no 
longer serves the purpose of a reference mark for observational measure- 
ments. 

We shall call the three constituents of any problem (a) the frame, (6) the 
comparison fluid, (c) the object system. 

In future developments frequent reference will be made to the com- 
parison fluid.t It must be understood that the comparison fluid is not a 
hypothesis. Still less is it an ascertained feature of the universe. The com- 
parison fluid is a datum of the elementary problems which we treat like 
the "frictionless constraints" so often specified as data in problems in 
elementary mechanics. There is no suggestion that a comparison aether 
actually exists throughout the universe. But, instead of a comparison 
aether, irregular distributions of matter exist and furnish landmarks for 
our observational measurements; the comparison fluid takes the place of 
these in our elementary equations. From these we can, if necessary, proceed 
to more complicated equations which take account of the irregularity of the 
background of our actual experiments. 

If anyone is disposed to offer criticism of this view of the equations of 
wave mechanics, it is necessary that he should state what (in his view) is the 
physical reference object intended by writers on wave mechanics when they 
mention positions, momenta, etc. For, if he believes the quantum equations 
to be true observationally, he must surely be prepared to state what observ- 
able system he believes them to be true of. We would remind him that a 

t The " standard environment*' occasionally referred to in earlier chapters is now replaced 
by the comparison fluid. 



11-2] The Eiemann-Christoffel Tensor 181 

geometrical frame is not observable; nor is it a simplified representation 
of anything observable ; its sharpness is, according to the principles of wave 
mechanics, incompatible with observability. 

The axiom of relativity is that we can only observe relations between 
physical entities. A relation implies two systems or two parts of a system; 
it can therefore occur only in connection with a double wave function, the 
duplicity corresponding to the two ends of the relation. 

It is therefore evident that the union of wave mechanics with relativity 
theory must be based on the theory of double wave tensors. As we pointed 
out in 10-75, the recognition that the simple wave tensor is an abstraction, 
and that the double wave tensor is the primary physical concept, is the 
counterpart in relativistic wave mechanics of the recognition in macro- 
scopic relativity theory that the interval is the primary physical concept. 

This conclusion is emphasised by the fact that the fundamental tensors in 
relativity theory are symmetrical tensors of the second rank, namely, the 
metrical tensor g^ v and the energy tensor T^ v . In symbolic notation these 
yield double wave tensors ^g^E^F^ and SIJ^JB^JJ,. On account of its 
antisymmetrical properties the Riemann-Christoffel tensor (although it is 
of the fourth rank) can also be expressed as a double wave tensor 



In the more elementary problems, the double wave tensors refer to the 
object system and comparison fluid, and specify the probability distribution 
of their combined configurations. What are ordinarily described as pro- 
perties of the object system must be understood to mean properties of the 
combination object system 4- comparison fluid. 

11*2. Linked Rotations. 

In general relativity theory, a vector A^ carried by parallel displacement 
round the perimeter of a surface element d8 va receives an incrementf 

dA^V^&lB^A'dS", (11-21) 

where B is the Riemann-Christoffel tensor. We have changed the usual 
order of writing the suffixes (which would have been B^ VOf ), since it is found 
to be very inconvenient in the present theory. We shall adopt local Galilean 
coordinates (natural coordinates); since real time is used, it is necessary to 
pay attention to the upper and lower positions of the suffixes. 

Let dS va be a circle of infinitesimal radius r in the plane x v x a , described 
in the direction x v ->x a . A vector A along the x axis, by receiving an incre- 
ment dAp along the x^ axis, is rotated in the x^x plane through an angle 



so that by (11-21) ^/= -iir 1 ^^. (11-22) 

t Mathematical Theory of Relativity, equation (84-1). 



182 Physical Applications [11-2 

The factor \ has disappeared because the surface element has two com- 
ponents dS va = -d/S ff " = 7rr 2 . 

The components of B therefore specify a linkage of two rotations, the one 
in the va plane and the other in the /*e plane, the first being a rotation of the 
position vector of the point considered and the second being a rotation of an 
arbitrary vector situated at the point considered. In physical applications 
we regard the point as occupied by an object system, and the rotations refer 
respectively to the position vector of the object system and to a vector 
regarded as intrinsic in the object system. We shall call the two vectors the 
position vector and the intrinsic vector. 

In ( 11-22) the angle of rotation of the position vector is va = 27T. The 
ratio of the linked rotations is therefore 



Or, expressing both angles contravariantly, 

0> /0"-= - \r*Bt" va . (1 1-232) 

In " de Sitter space-time ", i.e. space-time spherical in its space dimensions 
and hyperbolic in the time dimension, the Ricmann-Christoffel tensor is 

^%a = (W-S a MS/)/7? 2 . (11-233) 

This vanishes unless the planes /xc, va coincide. If /*, e = v, a the value is 
1/jR 2 . Hence, in de Sitter space-time the two rotations are in the same 
plane, and their ratio is 

e' vff /0 Vff = - ir 2 /JR 2 , (11-234) 

r being the length of the position vector (treated as infinitesimal) and E the 
radius of space-time. 

An illuminating point of view is obtained by interpreting 0' as the rotation 
of a comparison fluid. The intrinsic vector A^ has been parallelly displaced; 
this implies that, although its mathematical specification is altered, it has 
in some physical sense suffered no real change. It is still open to us to define 
the observational significance of the concept "real change"; this definition 
will decide the way in which the analytical theory of parallel displacement 
is to be utilised in the study of physical phenomena. We shall define real 
change to mean observable change. The orientation of the vector A in the 
geometrical frame is not observable; what we observe is its orientation in 
the comparison fluid which is the idealised substitute for material landmarks. 
Thus Ap will suffer no observable change if its direction remains fixed in the 
comparison fluid. Being fixed in the fluid it undergoes the same rotation 0' 
as the comparison fluid relative to the (unobservable) frame. 

The linkage of a rotation 0' of the comparison fluid to a rotation of the 
position vector of the object system is conceived dynamically as a recoil of 
the comparison fluid. If the fluid is assigned a moment of inertia 2w-ff 2 , 



11-2] The Riemann-Christoffel Tensor 183 

(11-234) expresses that when the object system is given an angular momen- 
tum mr^ddfdt the comparison fluid recoils with equal and opposite angular 
momentum 2mR 2 d6' jdt. In non-uniform space-time the comparison fluid 
has (as we should expect) a momental ellipsoid with unequal axes, so that 
the recoil angular velocity dO'/dt is not necessarily in the same plane as the 
recoil angular momentum, and therefore not necessarily in the same plane 
as de/dt. Thus components of ^ va for which /z, c ^ v, a are introduced. But 
we need not unduly stress this dynamical model. The R.C. tensor specifies 
directly the kinematical recoil of the comparison fluid corresponding to 
every possible cyclic displacement of the object system. If we introduce 
dynamical conceptions, we must attribute to the comparison fluid whatever 
dynamical characteristics are necessary to satisfy this specification, whether 
they are illustrated by familiar dynamical models or not. 

It may be noticed that a completely arbitrary connection between 0^' 
and v<r cannot be represented by an R.C. tensor; the arbitrary connection 
would have 36 disposable constants, whereas the R.C. tensor has 20 in- 
dependent components, t 

From the standpoint of macroscopic theory the foregoing is a highly 
speculative interpretation of the R.C. tensor. There is in fact an insuperable 
objection to adopting it in macroscopic relativity theory, which will be 
explained in 11 '3. But in microscopic theory we approach it in a different 
way; the speculative taint is removed; and the aforementioned objection 
disappears as soon as we substitute displacement by wave propagation for 
the classical conception of displacement. 

In wave mechanics the comparison fluid presents itself as a datum of the 
problem; without it the mathematical equations could have no relation to 
observable phenomena. We observe displacements of the object system 
relative to the comparison fluid; but the methods of mathematical physics 
require that we should use in our equations displacements relative to a 
geometrical frame. For convenience we call displacements relative to the 
frame absolute (i.e. conceived as absolute). We have therefore to analyse an 
observed relative displacement into absolute displacements of the object 
system and comparison fluid, before we can apply the theoretical equations. 
This partition can only be decided by convention. The convention is arbi- 
trary in the first place; but it may later be limited so as to fulfil conditions 
which simplify the resulting formulae. 

We have therefore to draw up a scheme of partition specifying the two 
absolute motions which correspond to any given relative motion; or equi- 
valently the scheme will specify the absolute motion of the comparison fluid 
corresponding to any given absolute motion of the object system. We have 

f We confine ourselves to Riemannian geometry. In affine geometry a generalised cur- 
vature tensor with 36 independent constants is substituted. 



184 Physical Applications [11-2 

seen that (for rotational displacements) such a scheme of linkage can be 
expressed by a R.C. tensor B^ V<r . 

Defined in this way as a scheme of partition, B^ va has no immediate 
relevance to curvature of space. But the analysis at the beginning of this 
section shows that, if we ascribe to the unobservable frame a metric whose 
B.C. tensor is B ftt909 the partition represented by B^^ becomes automatic 
on the understanding that displacement without observable change is 
represented geometrically by parallel displacement. "Observable", of 
course, means observable under the ideal conditions of the problem, namely 
that the only physical systems present are the object system and the 
comparison fluid. 

This double aspect of B^ va is the most essential link between relativity 
theory and wave mechanics. By it energy, momentum and mass, which 
appear in relativity theory as components of curvature of space-time, 
become identified with coefficients involved in the partition of relative 
displacements into absolute displacements, and therefore in the analysis of 
the double wave functions (which contain observational information) into 
the abstract simple wave functions which form the ordinary starting point 
of wave mechanics. In this way energy, momentum and mass get their 
footing in wave mechanics. 

The double interpretation of B^ va also exhibits the way in which the 
geometrical conception of mass as curvature becomes translated into a 
dynamical conception. The coefficients of the scheme of partition determine 
the recoil of the comparison fluid. An object particle whose displacements 
produce a large recoil is regarded dynamically as having a large mass. We 
see from.( 1 1-231) that if the B.C. tensor, and therefore the mass, is multiplied 
by any factor the recoil is multiplied by the same factor. 

11*3. Displacement by Wave Propagation. 

In obtaining (11-231) we assumed that the change dA^ recognised at the 
end of a complete cycle had occurred evenly during the cycle. In macroscopic 
physics it is impossible to admit this. When the object system receives a 
displacement dx , we do not know whether the infinitesimal displacement 
is going to form part of a circuit in the x^x^ plane or the x^x 3 plane, and it is 
therefore impossible to assign the plane of the recoil dO' of the comparison 
fluid. General relativity theory has therefore adopted a different represen- 
tation. It admits a rotation (usually accompanied by strain) of a vector 
parallelly displaced through dx\ but the rotation is described by a non- 
tensor quantity a 3-index symbol. The change assigned is thus dependent 
on the coordinate system employed; it cannot be pictured absolutely, or 
attributed to an objective rotation of the comparison fluid. 

This difficulty does not arise in wave mechanics, because we do not con- 



11-3] The Riemann-Christoffel Tensor 185 

template displacements of the classical type. The nearest approximation to 
classical displacement is propagation of a wave packet. But the wave packet 
is analysed into elementary waves, each of which fills the whole of space. 
An elementary displacement corresponds to the propagation of one of these 
waves. To each elementary wave there is a corresponding recoil of the com- 
parison fluid, and the recoil due to the propagation of the wave packet is the 
resultant of these. 

Wave analysis removes all ambiguity as to the plane of rotation, because 
it is a wave front, not a point, which is displaced. The free motion of a particle 
or wave packet is commonly analysed into "infinite plane waves " ; but these 
must be modified to fit finite space. In spherical space-time we can mark the 
wave front on the sphere, and obtain the wave propagation by rotating the 
sphere in the plane defined by the wave normal and the radius of curvature 
of space-time. For particles bound in an atom the analysis into elementary 
states gives waves representing angular motion; these correspond to 
rotations in three-dimensional space. 

It would be difficult to extend to all kinds of irregular space the con- 
ception of wave propagation as a simple rotation. In general the normal to 
space-time is six-dimensional. But wave mechanics evades all such com- 
plications by analysing physical systems into steady states in which the 
motion is characterised by constant angular momentum. Changes of 
distribution which cannot be represented by rotation (in the most generalised 
sense, i.e. displacement without intrinsic change in any of the planes of the 
symbolic frame) are treated by perturbation methods, and are not recognised 
as spatio-temporal displacements. If wave mechanics is unable to describe 
the change from configuration A to configuration B in terms of rotation, it 
represents the change as spatially discontinuous; that is to say, it examines 
the transfer of probability from A to B without representing the system as 
having passed through configurations intermediate between A and B. 
Such changes are called transitions, not displacements. 

The result is that every spatio-temporal displacement contemplated in 
wave mechanics is a rotation, or is analysed into rotations. Indeed the theory 
developed in the previous chapters makes no provision for displacements 
which are not the manifestation of rotation; and every continuous trans- 
formation or change of configuration is associated with a symbol defining 
a plane in a symbolic frame. Since we always know the plane of rotation 
associated with a displacement, the difficulty encountered in macroscopic 
physics does not arise. 

The obvious example in microscopic physics of an object system describing 
a small circle dS va is furnished by the electron in a hydrogen atom. We can 
contemplate the displacement of the electron during a time short compared 
with the time of revolution; but the displacement can only be described by 



186 Physical Applications [11-3 

an angle d0 va , not by a line element dx v , because the position angle of the 
electron at any instant is indeterminate. 

If then we consider an electron with constant angular momentum in the 
plane va 9 



the recoil of the comparison fluid in the plane vo will be, by (11-232), 



dt 
or, assuming spherical space, 

dg/ _ nh m.W 

"*""SS3P' (1133) 

It is desirable to meet at this stage the criticism that an effect of the order 
(11-33) is utterly trivial. The recoil dO'/dt applies to a comparison fluid 
filling the whole universe. There are, say, N other electrons in the universe, 
whose contributions to the total recoil must be added together. If (as seems 
to be generally implied in the elementary formulae) these are all supposed to 
be doing the same thing, the total recoil is 



where R' = R/VN-3 . 10" 13 cm. (11-35) 

according to the value of N found in Chapter xiv. This is by no means 
trivial; and outside the region of the nucleus (r > 10~ 12 cm.), dO'/dtis greater 
than d6/dt. 

If, on the other hand, we suppose that (11-31) refers to one electron only, 
it is right that the recoil (11-33) should be of the same order of observational 
insignificance as the motion which occasions it. To detect that one of the 
N electrons in the universe has acquired a rotation dO/dt can scarcely be said 
to be within reach of practical observation. It is true that we might observe 
a reasonably large effect, but the chances are N to 1 that we shall observe 
nothing; for there is nothing in the mathematical formulation to indicate 
that the electron referred to in (11-31) is the one which the observer is 
watching. 

In natural coordinates the continuum of space and imaginary time is 
isotropic. We have found that, for a spatial displacement of the object 
system, the comparison fluid recoils in the opposite direction; and the same 
holds for an imaginary time displacement. If we substitute real time, the 
sign of the corresponding components of the B.C. tensor is reversed; and the 
real time displacement of the comparison fluid is in the same direction as 
that of the object system. In our ordinary outlook, objects travel forward in 
time together. More particularly in a measurement involving an object 



11-4] The Riemann-Christoffel Tensor 187 

system and physical reference system, it is implied that both systems are 
observed simultaneously in the time reckoning adopted. We have seen that 
the scheme of linkage of displacements is initially an arbitrary convention, 
and for the present we shall treat it as arbitrary; but at the appropriate stage 
we shall specialise it so that the time displacements (in the time reckoning 
adopted) of the object system and comparison fluid are equal, as the 
ordinary outlook assumes. 

11-4. The Riemann-Christoffel Matrix. 

We shall now express the R.C. tensor as a wave tensor. In order to utilise 
the relation between wave tensors and space tensors found in our previous 
work, we must take the fourth coordinate to be imaginary time. Accordingly 
we use local rectangular coordinates with g^ = 8^"; and there is no longer 
any distinction between B* va and i^ tev<r . 

We define the Riemann-Christoffel Matrix to bef 



)y8- (11-41) 

Since the suffixes of the ^/-symbols run from to 5, we require components 
of Bp va additional to those which make up the ordinary R.C. tensor. The 
physical interpretation of the extra components will be found in due course; 
but their primary significance is that they specify linkages of the rotations 
of the object system and comparison fluid in the corresponding planes as 
in 11-2. 

Consider a factorisable matrix (as in (10-413)) 



Treating U and F as double vectors in the CD frame, we have by (10-24) 



By the symmetry property of the R.C. tensor, B^^^B^^. It follows 
from (11-41) that (-B) yj 3s=() y Sj8> so that, by (11-421), 

ZVfc-V* 

or UV =UV. This is satisfied if U and V are both symmetrical or both anti- 
symmetrical. We shall assume for the present that both are symmetrical; 
the antisymmetrical case is treated in 11-5. 

There is a simple generalisation of the identity (5-41), which extends it to 
any two four-valued quantities 0, #, namely 

<75 

S 



t It will be seen that the suffixes are so arranged that the B.C. tensor is represented in a 
CD frame ( 10*4). This has a slight advantage in certain later developments. 



188 Physical Applications [11-4 

This is proved in the same way as (5-41 ),f and holds for the six pentads given 
by M = 0, 1, 2, 3, 4, 5. Let tf ay ==0 aXy + Xa y . (11-441) 

Then (11-43) becomes 

{S a (J^ a ) a j8 (^ta)yS ~~ (^ie)a]8 (^leJyS) ^08 = ^' ( * 1*442) 

Multiplying initially by F ay and using (11-422), we obtain 

2<7JW-i6i8 = 0- (11-451) 

Also, multiplying initially by V y (J^ v ) a , we obtain 

SA/ur^O 0^")- (11-452) 

This last reduction depends on the rule(^? ft> ,) a (JS? /[1(r ) a jg = (E va ) p when a^ v.The 
term for which a = v reduces to V y ( 1)^ (E^)^ U^ , which cancels the last 
term V y (iE^)^ (i) y U^ , owing to the symmetrical property of U and V. 
We now express the results (11-451), (11-452) in terms of the Einstein 
tensor O^ v and invariant 0. Since ^ MV = 8^, 

? 4-7? 1 

p3 T -"^4^4 > / 1 i 4 i \ 

j- (11-4O1J 

We use the above definition of G^ v to define new components (? 56 and G^QQ. 
Then (11-451) becomes r 



n 1- 

V A A 

or, by the antisymmetrical properties oFB llVa9 

- G/iii + 5 5/i5,* + ^Ofio^t = ^leie (1 1-463) 

Summing (11-463) for /*,= 1, 2, 3, 4, we obtain 

0-055-0 w = 4fl 1616 . (11-464) 

But by (11-462) 



Hence, writing 6060 = JS 0505 = A (11-466) 

we obtain from (11-464) and (11-465) 

*i6i6 = !(#-2A). (11-47) 

Thus (11-451) and (11-452) become 



Whence, by (11-461), 

^v-^v(G f -2A) (11-48) 



by the usual relativity formula for the energy tensor T^ v . J 

t The reader who wishes to check the calculation may find it helpful to refer to a later 
result, (11-54), (11-551). 

J Mathematical Theory of Relativity, equation (54*71). The constant A is at our disposal, 
and its value must be fixed by a convention (see 13*1). Here the value of A is chosen 
so that the macroscopic energy tensor may have a simple connection with the energy, etc. 
defined in wave mechanics; it will, in fact, appear as we proceed, that the mechanical 
identifications adopted in current wave mechanics presuppose that energy, etc. are 



11-4] The Riemann-Christoffel Tensor 189 

If the axes 5 and are treated as invariant, (11-48) is a tensor equation, 
and therefore holds for any orientation of the axes in four dimensions. 

We have proved (11-48) for a factorisable R.C. matrix, but owing to the 
linearity of the equations the proof can be immediately extended to non- 
factorisable matrices, which must be expressed as the sum of a number of 
products UV in (11-421). As already stated, it is postulated that U and V 
are symmetrical matrices; when they are antisymmetrical some additional 
terms appear which will be investigated in 11-5. 

By (11-48) the energy tensor is composed of two parts, which we shall 
distinguish as a kinematical part (T^) k and an electrical part (T^ v ) e , namely, 



Considering the kinematical part, the component T^ v determines the rota- 
tion of the comparison fluid in the /x5 plane linked to a rotation of the position 
vector of the object system in the v5 plane. These are the rotations which 
correspond to translations in space-time (6-2). Accordingly: 

The kinematical energy tensor consists of those components of the whole 
Riemann-Christoffel tensor which specify linkages of translations in four 
dimensions. 

It is this part of the tensor which is used in dissecting a relative motion of 
translation into separate absolute motions of the object system and 
comparison fluid. 

We now see why this part of the R.C. tensor is sufficient for macroscopic 
mechanics. The " continuous matter" treated in macroscopic relativity 
theory is supposed to be such that.its kinematics can be described by macro- 
scopically continuous displacements in four dimensions. The theory makes 
110 provision for microscopic vorticity. Atomic physics, on the other hand, 
is intimately concerned with vorticity or angular momentum on a micro- 
scopic scale. Relativity theory can take account of microscopic motions 
(e.g. heat energy of a gas), but only as signless quantities mean-square 
values. If the atoms were spinning preponderantly in one direction, there 
would be no indication of this in the motions as averaged for macroscopic 
purposes. 

Thus the mechanics of the atom cannot be treated on the basis of the 
energy tensor T^ v alone. The complete R.C. tensor B fAva9 including com- 
ponents with suffixes and 5, plays the part of a general mechanical tensor 
applicable to atomic as well as macroscopic problems. The macroscopic 
energy tensor forms part of it, B^ v6 . In atomic problems it is generally 



measured from the zero defined by the values of A given here and in 11*5. In cosmical 
problems the value is fixed by another independent convention which has secured general 
recognition; this cosmical value will not necessarily agree with the value here used, as we 
shall find in (11-592) and (11-593). 



190 Physical Applications [11-4 

more convenient to handle the complete tensor in its wave tensor form, 
i.e. as the R.C. matrix (B)^^. 

It has always been difficult to see the physical significance of the process 
of contraction used in obtaining the energy tensor from the B.C. tensor. 
We have no physical insight into what we are doing when we add together 
^1212 + ^1313 + ^1414 * f rm ~~ ^11 We now see that from the physical point 
of view the energy tensor T n is not specially related to this sum, but is 
another component #1515 of the same tensor. The summation is part (but 
only part) of a process of calculating T u when other components are known 
depending on an identity satisfied by the components. But equally jB 1212 could 
be calculated from the same identity if the other components were known. 

In the generalised field theoryf I have derived the fundamental tensor 
*Bp Va from the theory of relation structure. Since only a portion of the 
tensor (viz. the part preserved in the contracted tensor *O^ V ) was actually 
used in specifying the electromagnetic-gravitational field, we seemed to 
have " dragged up from below a certain amount of apparently useless 
lumber" (loc. cit. 99). It was natural to hope that the unused part might 
ultimately be needed in the representation of microscopic structure; 
though at the time 110 progress could be made in this direction. We now see 
that this hope is fulfilled, and the full tensor *-B /4W e is utilised in microscopic 
theory. (*/^ lV(7 differs from B^ va in providing for a macroscopic electro- 
magnetic field, which would be an unnecessary complication at this stage 
of our investigation.) There is accordingly thorough continuity between 
the present theory and the field theory developed in my earlier book. 

Field theories which are based on "five-dimensional relativity", i.e. 
theories which introduce a curvature tensor in five dimensions, have not 
so simple a connection with the present theory. B^ va has the double 
aspect of a partition tensor and a curvature tensor; as a partition tensor 
it includes components corresponding to two additional suffixes 5, 0; but 
the identification with a curvature tensor applies only to the suffixes 
1, 2, 3, 4, and in the present theory we do not extend this identification. 
For such an extension it would be necessary to introduce a fictitious 
extension of the distribution over a fifth or sixth dimension, so as to 
provide corresponding Christoffel brackets and components of curvature. 
Even if this were formally possible, it would be a retrograde development, 
undoing the advance made in 6-3, 6-4. Whether the number of suffixes 
is four or six, B^ va is a function of four coordinates only. It is to be 
remembered that wave mechanics is a statistical theory. If we introduce 
fictitious dimensions, the generalised formulae will determine steady states 
in five or six dimensions ; but these are not the states with which we are 
concerned in physics. 

t Mathematical Theory of Relativity, Chapter vn, Pt. II. 



11-4] The Riemann-Christoffel Tensor 191 

As an elementary example, let the object system be a particle with an 
exact momentum. Let us choose the time direction so that it is at rest. Its 
energy tensor then reduces to a single component T^p^ where /> is the 
proper density of its probability distribution. By (11-231) a time displace- 
ment 80 46 of the object particle corresponds to a time displacement 80 45 ' of 
the comparison fluid, given by 



S0 4 5. (11-495) 

For any other direction of displacement SO' = 0. 

It may seem surprising that in this case the displacement of the particle 
in spatial directions causes no recoil of the comparison fluid. But it must be 
remembered that the conditions are highly idealised. By taking the momen- 
tum vector to be exact the position of the particle becomes entirely unobaerv- 
able. Therefore the problem of analysing an observed spatial displacement 
into absolute displacements of the object particle and comparison fluid does 
not arise. To represent observed position in space we must introduce a wave 
packet; the momentum then ceases to be wholly in one direction, and we 
cannot choose the time direction so that T^ v reduces to a single component. 
There will accordingly be components T n , etc., specifying the way in which 
the observed spatial displacement is to be partitioned. 

Returning to a particle with an exact momentum vector, the particle is 
distributed with even probability over the wave front of its waves. Dis- 
placement in the wave front is merely a transfer of our attention from one 
point to another, which has no dynamical reaction on the comparison fluid. 
Displacement normal to the wave front (relative to the frame) is a physical 
change of the conditions, involving a linked displacement of the comparison 
fluid. We see from (11-495) that, when the object particle moves forward in 
time, the comparison fluid moves in the same direction. We have pointed 
out that the usual outlook, which has become incorporated in the equations 
of physics, requires that an object system and its physical reference objects 
should always move together in time that the reference system should be 
a simultaneous one.f We have therefore the condition 

.ffS0 45 ' = r80 45 , (11-496) 

where J280 45 ' is the displacement of the comparison fluid in linear measure. 
By (11-495) and (11-496) ^ y^ Er (u . 497) 

When we said above that a time displacement of the particle was a physical 
change of the conditions, we regarded the displacement as being applied to 

f This refers primarily to coordinate time, and so depends on the choice of coordinates. 
But, in the present simple example, if the two systems agree in coordinate time they will 
agree in proper time. 



192 Physical Applications [11-4 

the particle and not to anything else just as one regards space displace- 
ments. But it is a rooted habit of thought that a time displacement applied 
to one particle automatically applies to the rest of the universe so that 
time displacement also is a mere transfer of our attention from the present 
to a future moment. We here reach the same outlook in a more legitimate 
way by so choosing the linkage of displacement that the object particle 
automatically carries the comparison fluid with it in the time direction. 

As another elementary example, suppose that the object system is 
identical with the comparison fluid; so that the object is its own standard of 
reference. Since the object cannot change relatively to its simultaneous self, 
it will be described observationally as uniform and unchanging. But a 
uniform unchanging distribution of matter is an Einstein universe. Setting 
80 45 ' = 80 45 in (11-495) to express that the object is always compared with 
its simultaneous self, we have =1/4777-2 (11-498) 

which is a well-known formula for the density in an Einstein universe of 
radius r. 

1 1-5. The Dual Riemann-Christoffel Tensor. 

By 10-4 the R.C. matrix can be resolved in alternative ways. Let 

(*)^ = Sfi^(^ (11-51) 

or, in the notation of 10-4, 

^. (11-52) 



We call b^ va the dual R.C. tensor. 

An interesting case is when the dual B.C. tensor consists of a single 
component 6 1616 = 6. Then, by (10-494), 

(B) = bE l sF u =bX(C D ) 9 (11-531) 

the minus sign applying to time-like matrices. Hence, by (11-52), 



the minus sign applying when E^ is time-like. 

Our first impression is that (11-532) is the R.C. tensor for de Sitter space- 
time (11-233), whose quadratic curvature Jt~ 2 is positive for space dimen- 
sions and negative for time dimensions. But in our present coordinate 
system a: 4 is imaginary time; the ambiguity of sign in (11-532) is eliminated 
when we substitute real time rotations for imaginary time rotations; so that 
(11-532) represents a world completely isotropic in space and real time. 

De Sitter space-time is obtained by taking the dual R.C. tensor to consist 
of a single component 64545=6; so that 

(B) 6^45^45 - J6S ( CyU (1 1-533) 

the minus sign applying to a = 01, 02, 03, 04, 05, 16 by (10-495). The plus 
sign applies to all the components of the ordinary four-dimensional R.C. 



11-5] The jRiemann-Christojfel Tensor 193 

tensor; these accordingly agree with the values for a de Sitter space-time of 
radius given by J6 = -R~ 2 . 

By (11-421), (11-51) and (11-533) 



Thus U and V in this case consist of single components ^45^45, $45^45, 
(^45^45 = b). Since J^ 45 is an antisymmetrical matrix, U and F are anti- 
symmetrical. 

Before proceeding further with the investigation of de Sitter space-time, 
it is necessary to extend the theory given in 11-4 which treats only the 
symmetrical case. Let 



According to (11-43) Q OLY = # ya . Using the standard matrices (3-27) for 
the pentad E 0a , we find by direct calculation that |(? ay is the matrix 

p"* &sXi-<AiX2 ^2X4-^4X2 ^4X1- ^1X4 

* ^1X2-^2X1 ^3X2-^2X3 0iX3-0sXi 

^4X2-^2X4 ^2X3-^3X2 ^4X3-^3X4 



This is equivalent to 



where ?7 = -- Since 



the result is iQ ay =^ ay -H^45)cjf 7 cJ-(^45)a y - (11-551) 

If T is another antisymmetrical matrix, we obtain a corresponding R.C. 

matrix ( J B).^-C^^W. X ^+ K^A-^o)^^. (H'M2) 

Since ?7 and F are both antisymmetrical, (5) ay 0s ^ s antisymmetrical in a 
and y and in j8 and 8. Hence, by (11-51), & Mwr = unless jE7 Me and JSf w are both 
time-like. The dual R.C. tensor thus has at most 36 components. We have 

U ay F ay = S b uva (E^ (E va ) ay = S b^ spur (JE^ J w ) 

= 426^^ = 47, (11-553) 

where 7 = 6 2 323 + 6 3isi + ^1212 + ^0404 + ^osos + ^4545 (1 1-554) 

Also we have 

) e U l (^ 46 ) ay F ay = S b vuno spur ( J 45 ^ ) spur ( 



Hence, setting j8,8 = a,y in (11-552), 

(11-56) 



194 Physical Applications [11*6 

The term 6 4646 is distinguished by the fact that J0 46 is the product of the two 
time-like matrices in the pentad E 0a . This rule enables us to write down the 
corresponding term for other pentads. 

The left-hand side of (11-56) is the quantity which was treated in 11-4, 
yielding the equation (11-451). In the symmetrical case it vanishes identic- 
ally. The right-hand side of (11-56) therefore gives the additional terms 
which must be included in (11-451) when U and V are antisymmetrical. In 
place of (11-462), we have the following equations for the diagonal com- 

ponenteof^ -0 11+ 1616 + 1010 +2& MM = 5 1616 +r (11-57) 

~ #22 + ^2625 + -B2020 + 2&3131 - JS 1616 + 7 

- <?33 + fi 3536 + -B 3030 + 26 1212 = B uu + T 
+ 4049 + 26 0806 = B uu + Y 

+ 2&0404 - #1616 + 7 
-G w + B OS06 + 26 4546 = 5 16ie + 7 

Adding the first four and subtracting the last two, we have 



So that, setting 5 1616 + F = i (0-2A), (11-581) 

we obtain A = # 0505 + 26 0404 + 26 464B - Y. (11-582) 

The result differs from the symmetrical case in two ways: (1) the value of 
A is changed, and (2) the energy tensor G flv %g tJiV (G 2X) now has three 
constituents. If T^(T flv ) k + (T^ v ) e + (T^) a9 (11-583) 

the four diagonal elements of the new constituent are 

(^)a= ( - I/**) (&2323 , ftilll. &1212, &0505)- (H'584) 

The non-diagonal elements will contain the non-diagonal terms of b jtva . 

We return now to the dual R.C. tensor with a single component 
6 4545 = b = 4jR~ 2 , which gives de Sitter space-time. Then (11-584) vanishes, 
and the energy tensor consists of kinematical components #^5 and elec- 
trical components U^ovo as * n ^ e symmetrical case. Here we have B^^ = E~ 2 
and B^Q= - JR 2 , by (11-533); so that T^ v = 0, as it should be in a de Sitter 
"empty" world. By (11-582), A = 35~ 2 , which agrees with the usual result.f 

The "emptiness" of the de Sitter world is here represented as due to a 
cancelling of two energy tensors (Tp V ) k and (T^. This is because the sim- 
plest kind of space is an electrically saturated space. To obtain a metric 
agreeing with that of neutral space we have to cancel the electrical energy. 
It is therefore appropriate that the formulae should exhibit the zero energy 
as the result of cancellation. Another way of obtaining the de Sitter world is 
to take a dual R.C. tensor with two components 6 4546 = 6 0404 = 6. Since the 
suffixes and 5 are interchanged in the two components, the four-dimen- 

t Mathematical Theory of Relativity, equation (69-12). 



11-6] The Riemann-Christoffel Temor 195 

sional part of B /[4eva is unaltered; but the energies B^^ 9 B^^ are found to 
be zero. 

An Einstein world is obtained by taking a dual R.C. tensor with two 

components 64545 = &0505 = 2 ^" 2 - (11-591) 

We then find # 2323 , -B 3 m B m2> S i6i5> B *m> ^3535= B ~ 2 ' 

#0404* ^1618 = ~^~ 2 - 

The other components vanish. These agree with the values in a space-time 
spherical in three dimensions and cylindrical in the fourth, namely /? 2 323> 

^3131 ^1212 =^~ 2 J -Bl414 ^2424 > ^3434 0- ^ Or ^ e ener gy tenSOr W6 obtalll 

(remembering that there is a component (244)0= (V^^osos) 

87ry^ = (-l, -1, ~1, -3)B-a. (11-592) 

By (11-582), A = 0. This is a possible specification of an Einstein world, 
though it is not the one which has usually been given in general relativity 
theory. The energy tensor ordinarily adopted isf 

' = (0, 0, 0, - 2) JB-2 (11-593) 



where A' = R~ 2 . That is to say, ( 1 1-593) is obtained from ( 1 1-592) by changing 
the constant A (the cosmical constant) in the formula 



In other words we reckon energy and pressure from a different zero. This 
change of zero reckoning will be explained in 14-3. 

11-6. Metric and " a priori" Probability. 

Let fcr^Jo- -(fl^- ty.,0), (H-611) 

87r(^ v ) c = A^ v , (11-612) 

so that ^ = (^)o-(^)c- (11-613) 

We shall suppose tentatively that (T^ and (T^ are the absolute energy 
tensors of the object system and comparison fluid referred to the frame. J 
The ordinary relative energy tensor T^ v of the object system is thus the 
difference of the two absolute tensors. 

In macroscopic relativity theory g^ v is defined by the condition that 
SpvdXpdXy ( =efo 2 ) is a quantity measurable in a definite way by scales and 
clocks. This definition is clearly unsuitable for microscopic theory. We must 
adopt a definition of g^ v in terms of more primitive conceptions, and show 
later, when the theory is sufficiently developed to treat the extension of 
macroscopic aggregations (scales and clocks), that the macroscopic definition 
is equivalent. We therefore regard (11-612) as the definition ofg^. 



t For real time the last component becomes 87rT 44 '= +2 R~* 9 giving p 

{ The absolute energy tensor referred to the frame (self -energy tensor) is defined in 10*5. 



196 Physical Applications [11-6 

Apart from a numerical scale factor A/STT, the metrical tensor g^ v is simply 
the energy tensor of the adopted comparison fluid. 

We have seen that the geometrical frame of coordinates must be filled 
with a comparison fluid in order to form an ideally observable background 
of reference for the object systems that we study. The comparison fluid 
consists of a probability distribution of particles which might be specified 
in detail; but macroscopically it is sufficiently described by specifying its 
energy tensor (relative to the frame) as a function of the coordinates. This 
energy tensor constitutes a metrical tensor for the frame and is more 
familiar to us under that name. Having thus defined g^ v9 we can transform 
the coordinates locally to rectangular coordinates. Then (11-612) becomes, 
on raising a suffix, (r/) c = (A/87r) S/. (11-614) 

That is to say, the comparison fluid turns out to be uniform and isotropic at 
every point so far as the gross characteristics described by the mixed 
energy tensor are concerned. We do not choose an isotropic comparison fluid . 
Whatever comparison fluid is chosen turns out to be isotropic, because, 
being the comparison fluid, it is the standard of isotropy . 

In elementary problems we consider only one or two particles. It is not 
intended that there shall be no other particles in the universe. It is implicitly 
assumed that there is some innocuous way of distributing the other particles 
so that the results of our investigation will not be entirely invalidated by 
their presence. Formally what is known as "the problem of two particles" 
should be described as the problem of N particles, of which two are studied 
in detail whilst the remaining N 2 are supposed to have some standard 
average distribution which remains undisturbed by the behaviour of the 
two particles. We shall call the N 2 particles unspecified particles. A par- 
ticle is specified by giving it a wave function describing a particular pro- 
bability distribution of position or momentum; so long as its probability 
distribution is the standard average distribution there is no need to mention 
it in the problem. 

It would be redundant to surround the object system with a double 
environment of (a) unspecified particles waiting to be introduced into the 
analysis as the treatment grows more comprehensive, and (6) the particles 
constituting the comparison fluid. We therefore adopt a comparison fluid 
consisting of all the unspecified particles in the universe. Let the number 
be N. It is convenient also to define a partial comparison fluid having 1/^th 
of the density of the actual comparison fluid, and therefore corresponding to 
the standard probability distribution of one unspecified particle . The absolute 
energy tensor of a partial comparison fluid is, by (11-612), 

(^-(A/SidOfiW- (11-615) 

Whilst it is permissible to speak of the energy tensor T^ v belonging to a 



11-6] The Riemann-Christoffel Tensor 197 

particular object particle, we must not use the formulae equating it to 
curvature of space-time, unless the object particle is the sole occupant of 
the region considered. Since the probability distributions of a number of 
object particles may overlap, we must write (11-611) more generally as 

87^(2^)0= -(G^-^,0). (n-621) 

Here S extends over all the particles in the universe treated in turn as 
object particles, although in practice the main contribution comes from a 
few particles specified as being present in the region we are considering. 
(The combined density in any region of all the unspecified particles is 
probably less than 10~ 28 gm. per cu. cm.) 

We have therefore for the whole energy tensor in any region 

2 I ^=S(r^-(r M>p ) e L{(2'^-(2' M ,y, (11-622) 

there being N terms in the summation. Thus each object particle has the 
relative energy tensor (T ^. (T ^. (11 . 623) 

When an object particle is unspecified, it has the standard probability 
distribution, and its absolute energy tensor (IJ^Jo is equal to (T^ . Thus 
its relative energy tensor vanishes. It follows that for a specified object 
particle: 

The relative (observable) energy tensor is measured from the standard 
probability distribution as zero, and not from a hypothetical state of noii- 
existence. The conception of creating a particle does not enter into our 
theory. Energy is furnished to a particle by specifying it, not by creating it. 

The result (11-623) is the justification of the tentative procedure which 
we have been following in this section. It is necessary in order to conform 
to the general outlook of wave mechanics, which allows us to discuss a 
system of a few particles without explicit recognition of the vast number of 
other particles in the universe. If T^ v were merely a tensor involved in the 
inner mechanics of the system, it would not matter much from what zero 
it was reckoned. But in unified theory T^ is to be identified with the 
relativity tensor (-l/87r){G [ flJ/ -^ lf (G-2A)}; and for this it is essential 
that all the particles in the region, specified or not, shall be included. If in 
the formulation of the " problem of two particles" it is intended that there 
shall be no unspecified particles overlapping the system, this should be 
stated explicitly; the appropriate terms must then be introduced into the 
equation to represent the energy employed in creating a vacant space 
among the JV - 2 unspecified particles; in fact the unspecified particles must 
be specified as absent from the region. Since this is not the procedure 
followed, we must adopt the alternative given by (1 1-623), namely we must 
choose the zero of energy in such a way that the unspecified particles make 
no contribution to 2^'BO that we may ignore them without harm. 



198 Physical Applications [11-6 

Our results up to this point are: 

(a) The adopted comparison fluid consists of the unspecified particles of 
the universe. 

(6) The absolute energy tensor of the comparison fluid is the A-term 
(generally known as the cosmical term) in the relative energy tensor. 

(c) The absolute energy tensor of the comparison fluid is the metrical 
tensor of the frame. 

If a different comparison fluid were used it would be necessary to 
"specify", either individually or macroscopically, the particles .composing 
it. The macroscopic reference objects used in particular observations can 
be regarded as a specified comparison fluid. But it is simpler to treat these 
as additional object systems; so that the same comparison fluid (a) is used 
as intermediary in all problems, and the observed relations between object 
systems are resolved into relations (unobserved, but nevertheless ideally 
observable) between each object system and a common comparison fluid. 

The probability distribution which applies in the absence of special 
information is commonly called the a priori probability distribution. The 
unspecified particles are those as to which we have no special information. 
The probability distribution of an unspecified particle, which we have 
identified with a partial comparison fluid, is therefore the a priori pro- 
bability distribution. The identification 

A priori probability distribution = partial comparison fluid 
helps to connect our theory with current terminology; but it must be under- 
stood that we reject emphatically the conception of a priori probability. 
The distribution is uniform and isotropic not for a priori reasons, but for 
the reason already given, namely that it determines the tensor g^ v . It is, 
I think, generally admitted that the need for a conception of a priori 
probability is a logical weakness in statistical mechanics as ordinarily 
developed. We need not discuss here attempts to defend it on a metaphysical 
basis, and to deduce from a ' ' Principle of Indifference ' ' that it has a uniform 
and isotropic distribution. Probably few theorists would accept such views 
today. But it is not generally realised that relativity theory has rendered 
the conception of a priori probability entirely unnecessary ,f 

The so-called a priori probability distribution used in statistical mechanics 
is essentially an unobservable; for as soon as we have any observational 
information, the a priori probability ceases to apply, and a modified (actual) 
probability is substituted. A priori probability is to be treated therefore as 
other unobservables (introduced for mathematical convenience) are treated 
in relativity theory, e.g. frames of space and time. The a priori probability 
distribution is sometimes called a "basis of statistics"; that is to say, it is 
f New Pathways in Science, pp. 129 ff. 



11-6] The Riemann-Christoffel Tensor 199 

a standard of reference for statistical enumerations in the same way that a 
frame of space and time is a standard of reference for measurements of 
extension. Observable results are invariant for all transformations of these 
reference frames; they must equally be invariant for changes of the adopted 
a priori probability distribution. The common appellation is therefore a 
misnomer; the a priori probability distribution is not anything that is 
given a priori; it is an arbitrary comparison distribution which we can change 
as we please. 

The reason why it is not ordinarily recognised that observable phenomena 
are invariant for transformations of the a priori probability distribution, is 
that the transformation is not pursued to the end. If the change of reference 
distribution changes the equations of an electron and proton, it will change 
also the equations determining the behaviour of the measuring appliances 
composed of electrons and protons. When the transformation is applied 
both to the measuring appliance and the object measured, the result of the 
measurement is invariant. But if we apply the transformation only to the 
microscopic systems which are being measured and not to the macroscopic 
measuring apparatus, we fail to notice this invariance. That was the mistake 
originally made in interpreting the Michelson-Morley experiment; the 
transformation representing change of motion was at first applied only to 
the optical factors in the experiment, the mechanical or metrical factors 
being overlooked. 

This invariance is provided for in the usual way by a tensor relation 
between the quantities concerned in the microscopic and macroscopic parts 
of the theory. In the microscopic equations the change of a priori pro- 
bability is formulated as a change of the energy tensor (3^^) c of the com- 
parison fluid; in the macroscopic equations (which determine the behaviour 
of the measuring appliances) the same change is expressed as a change of 
the metrical tensor g^ v . 

The linkage of the changes of g^ v and (T^ c , so as to produce no observable 
result, is provided for by their description as tensors of the same kind. In 
(11-612) we have gone further, and identified them. The importance of this 
step is that if the frame and the comparison fluid are both described by the 
same tensor called in one case the metrical tensor and the other case the 
energy tensor it minimises the error of current quantum theory in con- 
fusing them. To give meaning to the ordinary equations of current quantum 
theory we have to suppose that a particular physical reference object is 
intended, and we have to discover what this reference object is. If, on being 
challenged, the quantum physicist were to reply that he intended the 
reference object to be an atom of carbon travelling at 150 km. per sec. in 
the z-direction, it would be preposterously misleading not to have stated 
this. If, however, he replies that he has not distinguished between the frame 



200 Physical Applications [11-6 

and the comparison fluid, so that when he said that the tensor g^ v of the 
frame was isotropic (i.e. that his coordinates were rectangular) he also meant 
that the comparison fluid was isotropic, his omission is a minor inadvertence. 
He might perhaps accuse us of stupidity if we did not supply the omission 
for ourselves. Having regard to this, our identification of g^ v with the energy 
tensor of the comparison fluid is an obvious rectification of an omission in 
current quantum theory. 

Locally any change of (2^ v ) c or g^ v is a tensor transformation; but over an 
extended region we have to distinguish between tensor transformations 
and absolute changes. From the second derivatives of the g^ we obtain 
certain characteristics of the comparison fluid, which we shall call hyper- 
characteristics, which must be specified definitely in any problem and must 
not be changed in any transformation. These hypercharacteristics corre- 
spond to irreducible gravitational fields, or to the distributions of matter 
causing such fields. Either an assurance that the field is zero, or a specifica- 
tion of the invariants of the field, is an essential datum of any problem con- 
cerning an object system which covers an extended region. The observable 
results are not invariant for transformations of the probability distribution 
of the unspecified particles which would create a change of irreducible 
gravitational field; and when, as usual, the comparison fluid is not mentioned 
explicitly as a datum of the problem, it is obviously implied that its cha- 
racteristics are not incompatible with the other data furnished, that is to 
say with the invariants of the gravitational field. 

Accordingly the statement that any distribution may be chosen as an 
a priori probability distribution requires qualification when an extended 
region is considered. It must be remembered that we are not defending 
a priori probability. In our theory the partial comparison fluid takes its 
place. Therefore we are not led into a discussion as to what kind of field 
should be assumed as probable a priori, if no observational information is 
furnished. If we are not told what the field is, we have no idea what will 
happen, or what is likely to happen and there is no more to be said. 

By ( 1 1-623) the unspecified particles of the universe make no contribution 
to Tp V , and can therefore be neglected in elementary problems at any rate 
so far as this part of the theory is concerned. But there are other ways in 
which the treatment of a system of a few particles as if they were the only 
particles in the universe is liable to be misleading. It is natural to discuss 
the effect of spatial displacement of a single particle, the other particles in 
the universe being unchanged. It is possible also to discuss a temporal 
displacement of a single particle, the other particles being unchanged. This 
conception is used frequently and legitimately in relativity theory; indeed 
it would be contrary to the fundamental principle of relativity theory to 
discriminate between space and time displacements. But in quantum 



11-7] The Riemann-Christoffel Tensor 201 

physics the notion of temporal displacement has been employed less care- 
fully. Usually when reference is made to a displacement (da?, dy, dz, dt) of 
a particle, it seems to be intended that (dx, dy, dz) applies to that particle, 
and dt applies to all the particles in the universe. Obviously, when this 
interpretation is intended, Lorentz transformations are not applicable to 
(dx, dy, dz, dt). Thus the common form of the "problem of one particle" 
combines spatial displacement of one particle with temporal displacement 
of N particles. The "one" particle is thus a hybrid whose time displace- 
ments are N times as effective as its space displacements in producing recoil 
of the comparison fluid; and the corresponding Riemann-Christoffel tensor 
has its time component 4545 magnified N times relatively to its space 
components B 1515 , etc. The corresponding radius of curvature is therefore 
R in the space directions, but R' = R/^/N in the time direction. Since N is 
about 10 79 , the geometrical picture has become too extravagant to be 
retained; but we express the same analytical relation by ascribing to the 
intrinsic vectors of the object particle the appropriate rotation 80', linked 
by .64545 to the rotation 8tf R of its position vector in space-time. This very 
rapid rotation of the intrinsic vectors (e.g. the stream vector) is interpreted 
as a periodic wave. The wave length should be 27TJR', which is of order 
1C" 12 cm., and is comparable with the wave length of the de Broglie waves 
of elementary particles. The precise calculation of the wave lengths for 
protons and electrons involves additional considerations which will be 
treated in due course. 

The waves attributed to an electron in current wave mechanics belong to 
it, not individually, but because it is one of N particles; and their wave 
length is dependent on the number N. Considered by itself, a particle has 
no other periodicity than that due to the connectivity of the space in which 
it is situated. The wave length of its "waves" is the circumference of space. 

11-7. Microscopic and Macroscopic Theory. 

In the most general formulation of the problems of physics the particles are 
divided into three groups: 

(a) One or more particles specified individually. 

(6) Collections of particles specified macroscopically. 

(c) The "rest of the universe " consisting of unspecified particles. 

In mathematical treatment the subject is divided into a number of idealised 
problems in which one of the groups may be absent. 

In general the particles of group (c) far outnumber all the others, and most 
of our formulae are developed on this understanding. There is, however, a 
class of problems, namely, cosmical problems, in which the whole universe 
is specified macroscopically, e.g. an Einstein universe; there are then no 



202 Physical Applications [11-7 

particles of group (c), and special precautions must be taken in applying the 
formulae (Chapter xiv). 

By (11-623) the energy tensor of the unspecified particles vanishes. But 
there is no such condition for the other components of their B.C. matrix; 
and the four-dimensional R.C. tensor of the unspecified particles is the 
foundation of the R.C. tensor of space-time. We shall call this the cosmical 
E.G. tensor. If it exists alone it gives a de Sitter space-time; for that is well 
known to be the only solution (without singularities) of the equation 
a tlv -$g fJLV (G-2X) = satisfied by space-time containing only unspecified 
particles. 

Notwithstanding the overwhelming preponderance of unspecified par- 
ticles, the effect of group (6) is often more important. This is because the 
effect is more localised. As part of their macroscopic specification, the 
position of these collections of particles is closely defined; and they con- 
tribute an R.C. tensor which locally far outweighs the cosmical R.C. tensor. 
Consequently the latter is commonly neglected in the local problems of 
general relativity theory, e.g. the theory of planetary motion. 

Quantum theory deals primarily with problems involving (a) and (c) only. 
The influence of macroscopic objects, if any, on these problems is attributed 
to gravitational or electromagnetic fields emanating from them. We may 
therefore call problems which do not involve group (6), directly or indirectly, 
field-free problems. In field-free conditions space-time is of the de Sitter type. 
The specified particles (a) are represented by wave-functions, and the un- 
specified particles (c) are represented by the cosmical R.C. tensor, or 
equivalently by the g^ v which correspond to it. 

It might be thought that the individually specified particles, like those 
of group (6), would contribute a-local R.C. tensor very much greater than 
the cosmical tensor. For example, the average density in the space occupied 
by an atom is very much greater than the density ( ~ 10~ 28 ) due to un- 
specified particles. But this is a confusion of ideas. We must make up our 
minds whether the R.C. matrix of a particle is to be absorbed into the R.C. 
tensor of space-time, or reserved for more detailed treatment by quantum 
theory. We cannot have it both ways. A particle specified individually is one 
of those reserved for detailed treatment. Its interaction with its surround- 
ings is expressed by wave functions. We must not at the same time insert 
a duplicate in the R.C. tensor, and so obtain another (cruder) representation 
of its interaction by the changes of g^ v which would result. 

We shall find later that the wave functions of a specified particle, employed 
in current theory, are defined in such a way that they represent an addition 
to, not a substitution for, the characteristics belonging to it as an unspecified 
particle. In that sense the specified particle is represented in the R.C. tensor 
as well as in the wave functions. But that merely secures that the cosmical 



11-7] The Eiemann-Christoffel Tensor 203 

tensor is not disturbed when the particle is specified. If it were not for this 
provision, the R.C. tensor of space-time would be reduced by about 1 part 
in 10 79 for every particle specified. As it is, we may regard the number of 
unspecified particles as a fixed constant N, giving a constant B.C. tensor 
however many particles may be specified individually. 

It is important to realise that the individual atoms and electrons, whose 
energy and momentum are given by their wave functions, do not disturb 
the curvature of the space-time in which their wave functions are represented . 
Reciprocally the macroscopic objects, whose energy and momentum are 
represented by components of curvature of space-time, have no wave 
functions.f For the purpose of investigating the connection of microscopic 
and macroscopic theories, we can represent macroscopic objects by com- 
plicated wave functions; but such wave functions occupy a space-time which 
has no more than the cosmical curvature. In 11-1 we referred to the error 
of confusing the frame with the physical comparison system, and remarked 
that the neglect of uncertainty of position and momentum of light com- 
parison objects and the neglect of curvature of space-time produced by 
heavy comparison objects were different forms of the same error. The error 
takes one form or the other according as the comparison objects are specified 
microscopically or macroscopically. 

Using the notation of 10-4, the R.C. matrix can be expressed in the 
various forms: 

(11-711) 
(H-712) 
^(E^. (11-713) 

Let us consider in particular the cosmical R.C. matrix which gives a metric 
of the de Sitter form. By (11-533) this isj 

(B^^^E^F^B^^C.D^^C^}. (11-721) 
This is the space tensor; the corresponding strain tensor is 

(ntraln = 4tf~ 2 Vl6=^ (H'722) 

as in (10-54). By 10-5 this represents an "uncertain" momentum in the 
CD-frame, or an "exact" momei^tum in the JEJjF-frame. 

Consider first the GjD-frame. By (11-712) the R.C. tensor B^ vo , which 
constitutes the complete energy tensor, has been constructed in the CD- 
frame; and it would therefore seem that the uncertain energy tensor is the 
true cosmical energy tensor. But in that case the position tensor is exact; 

t Or we may use wave functions which represent the kinetic energy and pressure of the 
particles of a gas, but not their rest energy; in that case the curvature of the space-time in 
which these functions are represented corresponds to the rest energy, but not the kinetic 
energy. 

% The suffixes k, e, s, t refer to kinematical, electrical, space-like and time-like matrices. 



204 Physical Applications [ ' 1 1 7 

that is to say the cosmical R.C. tensor is wholly concentrated at one point 
of space. This, of course, does not correspond to the actual conditions of de 
Sitter space-time. The failure is easily explained; we cannot do more than 
we have done with one double frame. The cosmical R.C. tensor is made up 
of contributions from some 10 79 unspecified particles, each of which has its 
own double frame. If we attribute to them uncertain momenta, they will 
have exact positions; but between them they will cover all space, giving 
a practically continuous R.C. tensor. 

Since (-B) Btraln is algebraic in the J^JP-frame, we can write, instead of 
(11-722), 

(*)strain=~^-M4^^ 

(11-73) 

introducing N double frames instead of one. If the frames commute (Case (b) 
in 2-7) they have no relation of orientation to one another. The method 
of imposing a uniform distribution of orientation belongs to the type of 
treatment of the many particle problem introduced in Chapter xvi. Here it 
is sufficient to recognise that the difficulty of avoiding concentration of the 
R.C. tensor at one point will disappear when we take account of its highly 
composite nature. 

But we have a great deal more to learn from the double frame, before 
proceeding to the many particle problem. We consider the simplest of all 
problems an elementary object particle in de Sitter space-time. The un- 
specified particles, which provide the de Sitter metric, form the comparison 
fluid for the particle. To bring the problem within range of treatment by 
double wave functions, we use a partial comparison fluid consisting of one 
unspecified particle. The object particle, being specified microscopically, 
does not contribute to the curvature of the space-time in which we represent 
its wave function. The cosmical R.C. tensor at any point corresponds to a 
uniform distribution of momentum vectors in all directions. We could 
suppose that this was a superposition of exact momentum vectors of a 
great number of unspecified particles; but it is evidently more convenient 
to take it to be the entirely uncertain momentum of one unspecified 
particle a partial comparison fluid. This will have definite position in the 
small region we are considering; but the other unspecified particles will 
provide similar R.C. tensors at other points. (We take advantage of our 
liberty to choose any convenient way of analysing the unspecified matter of 
the universe into "particles", i.e. wave systems.) The cosmical R.C. tensor 
at the point is the complete self-energy tensor of the partial comparison fluid. 
We have represented it in a CD-frame. By transforming to the J57JP-frame 
(so that the R.C. tensor is transformed into its dual 6 /ACva ) we change it 
into an exact energy tensor 6^45^45. We then readily recognise it as the 



11-8] The Riemann-Christoffel Tensor 205 

outer square of the stream vector of a neutral particle which was identified 
with the self energy tensor in 10-5. The unspecified particle is neutral, 
for otherwise it would not give the R.C. tensor characteristic of neutral 
space-time; also when the "entire uncertainty" applies to all components 
of a complete stream vector the sign of the charge necessarily shares in the 
uncertainty. 

Turning to the elementary object particle, since we are considering only 
a point, or small region, we must for consistency suppose it to have exact 
position and uncertain momentum. This condition is not easy to formulate 
since we do not want the charge and spin to be uncertain. It is simplest to 
start with an elementary particle and its partial comparison fluid defined as 
having exact momentum in the UjF-frame. We can afterwards transpose 
the results to the CD-frame. The product of the stream vectors of the object 
particle and of the partial comparison fluid will give the mutual energy 
tensor. This when transposed to the 6 y D-frame will give the R.C. matrix of 
the object particle. The portion B iav ^ + B llXM will give the ordinary energy 
tensor of the object particle. This as we have seen is the observable relative 
energy. Thus the relative energy defined by T^ v is the same as the mutual 
energy defined in 1O5 as the product of the stream vectors; except that 
the latter is expressed as a complete tensor and exhibits the kinematical 
and electrical energies separately, whereas T^ v gives their sum. We can 
separate the part corresponding to the self energy of the comparison fluid, 
as in (11-613), the remainder being regarded as an energy of the object 
particle.t For consistency this should agree with the self energy of the 
object particle defined directly in (10-51). This condition is investigated 
in 12-6. 

1 1-8. Neutral Comparison Fluid. 

By working in the J5? JF-frame we are able to use particles with exact stream 
vectors. Consider an object particle with stream vector C7 = SJK /i % fi = ^x* 
and a comparison particle with stream vector F = SjK ft v ft = ^eo*. The com- 
bined system is described by double wave vectors Y, X, where 

T = ^*, X = xo>*. (11-81) 

The microscopic treatment will depend on the use of these wave functions ; 
but the connection with macroscopic theory is given by the identification 

t Since the object particle and comparison fluid move in opposite directions relative to 
the frame in our coordinates (space and imaginary time), the self energy of the partial com- 
parison fluid should be taken to be minus the square of the stream vector in order that it 
may be comparable with a mutual energy. An unspecified particle treated as object particle 
would produce a recoil of a similar unspecified particle treated as its partial comparison fluid. 
This explains why the total energy in (11-613) is the difference, not the sum, of the energies 
of the object particle and comparison fluid. 



206 Physical Applications [ 1 1- 8 

(comparing (11-711) and (11-713)) 



where 6 /iv(r is the dual R.C. tensor of the object particle with the other 
particle as comparison fluid, together with the condition that the dual 
cosmical R.C. tensor b jMtr f which determines the metric is given by 

V V " = *fitwr'- (11-822) 

We have been mainly using space tensors, because we wished to connect our 
results with the space tensors of general relativity theory; but for the 
special purposes of wave mechanics strain vectors are more useful. We shall 
therefore take $, x> #> <*> * be covariant wave vectors forming a strain 
tensor 6 jltW . 

In this simple formulation we have taken the comparison particle (or 
partial comparison fluid) to be an elementary charged particle. Our standard 
equations, however, postulate a neutral comparison fluid. We wish now to 
direct special attention to the step by which we pass from a charged particle 
to a neutral particle as comparison fluid. The relation between the stream 
strain vectors F, V$ of an elementary and a neutral particle is 

TJ = qsF (11-83) 

This follows at once from (6-64), the strain vector of the neutral particle 
(equally likely to be in any one of the four states) being J (8 a + 8 b + S C + S d ). 
We can therefore pass from an electrically saturated to a neutral com- 
parison fluid by contracting the suffixes of ^ and w. The change is expressed 
by the transformation . , , , , m j.\ 

7 f OL < rBXv* lt *8'~*'iTOtYBXv***B (JLJi'o*) 

or, in terms of wave functions, 

Y^X^JY^Xrf. (11-85) 

In (11-85) we need no longer assume that Y and X are factorisable. A state 
of a double system is regarded as elementary if it is pure as regards factorisa- 
tion into Y and X, irrespective of whether there is a UV factorisation. The 
formula (11-85) is of great importance in Chapter xn. 

We have at last reached the solution of what might be called the initial 
problem of electron and proton theory. The microscopic complete relative 
(or mutual) energy strain tensor of an elementary charged particle together 
with a neutral comparison fluid has the form Y a 0X y . As the importance of 
this may not be obvious, we make the following comments: 

(1) The elementary charged particle is usually described by simple wave 
functions ^, #, and the energy and momentum are contained in a simple 
stream vector 0x*- By introducing an empirical mass constant (different 
for the proton and electron) these simple wave functions are made to serve 
the ordinary purposes of quantum theory and yield correct results. But if 



11-8] The Riemann-Christoffel Tensor 207 

we are not content to accept the mass as an empirical characteristic peculiar 
to wave mechanics, and wish to trace its connection with the macroscopic 
mass identified with a component of space-time curvature, we are met with 
the difficulty that all the measurable characteristics of a charged particle 
are contained in double wave functions of the particle and a physical com- 
parison system. The conditions of replacement of the double wave functions 
by single wave functions is the central problem in the unification of rela- 
tivity theory and current quantum theory. We take a large step towards 
the solution when we find that, adopting a neutral comparison fluid, the 
double wave functions are of the specialised kind Y a p, X y p, the second 
suffixes being "contractible". 

(2) Unlike $ and #, the wave functions T, X are directly connected with 
the fundamental tensors of relativity theory. Taking the energy tensors of 
a number of particles (such that the components ignored in macroscopic 
theory cancel out) we can calculate the curvature of space-time equivalent 
to them, and hence translate the energy, etc., into macroscopic units. 

(3) The replacement of the energy tensor TX* by an energy vector ^x* 
occurs by contraction of the suffixes j8, 8, which is the usual method in tensor 
calculus of reducing the rank of a tensor. 

(4) We have taken the comparison fluid to consist of one neutral particle, 
that being all that we can provide for with one double frame. It is therefore 
a partial comparison fluid according to our previous terminology. The 
passage to the actual comparison fluid consisting of N unspecified particles 
is, however, quite simple. The analysis in (11-73) represents the energy 
tensor as the sum of independent tensors in different frames; so that each 
object particle affects only its own partial comparison fluid. The average 
recoil of the total comparison fluid is therefore 1 /Nth of the amount of the 
recoil of the partial comparison fluid. The Riemann-Christoffel tensor is 
thereby lessened in the ratio 1/N. 

(5) The ideal conditions postulated in these equations correspond to 
de Sitter space-time containing no macroscopically specified matter. It 
should be remembered that de Sitter space-time is, according to ordinary 
standards, an expanding universe. It will be necessary later to treat the 
modifications involved in using a static frame of reference (Einstein space- 
time) but this involves introducing macroscopically specified matter, 

We shall now show that the contractible double wave vectors Y a 0, X y p 
can (for practical purposes) be replaced by simple wave vectors a , x y The 
observational significance of wave functions is that they determine expecta- 
tion values for the system of operators P which represent observable 
relations. Ordinarily for double wave vectors (appropriately normalised) 
the expectation value is X a gP ayt pT y 8; but in the present application the 



208 Physical Applications [11-8 

contraction of the second suffixes restricts P to the form P ay , and the 
expectation value is X P *F (11'86) 

Introduce eight simple wave vectors 0#>, rf (j8= 1, 2, 3, 4) defined by 

W-Ytf. x.W-X.0. 
Then (11-86) becomes 

X. W ^y* y n) + X^ ( ll ' 87 > 

This is the expectation value of P with respect to an impure simple wave 
tensor 22) 33 4). ( 1 1-88) 



By using symbolic factors of T the expectation value may be written 
FPTasin (3-35). 

Thus, for the purpose of calculating expectation values, the pure (con- 
tractible) double tensor *FX* can be replaced by the impure simple tensor T. 
The reduction of double wave tensors to simple wave tensors is therefore 
legitimate, but involves a re-analysis into pure elementary states. 

We have considered the reduction of an isolated (discrete) wave tensor. 
There still remains the question of replacing the phase space of the double 
wave tensor by that of a simple wave tensor. We shall see in the 'next 
chapter that this transformation of the phase space has important con- 
sequences. 

11-9. Retrospect. 

In this chapter a gulf between macroscopic relativity theory arid micro- 
scopic quantum theory has been bridged. We have found how to construct 
a Riemann-Christoffel matrix, (a) out of the tensors employed in general 
relativity theory, and (b) out of the wave functions employed in quantum 
theory. The mechanical quantities, energy, momentum, etc. referred to in 
the two theories are thus reduced to a common form. We can, for example, 
add the R.C. matrices corresponding to the wave functions of a large 
number of particles and derive the energy tensor which describes the 
aggregation macroscopically. 

Whereas the construction of the R.C. matrix out of macroscopic tensors 
is straightforward, the theory of its construction out of wave functions is 
perhaps unexpectedly intricate. It is natural to put the question, What 
is the R.C. matrix corresponding to a proton or electron with an exact 
momentum vector? The question is not so elementary as it seems, because 
the properties ascribed to the particle are really combined properties of 
the particle and a standard comparison fluid; and although the particle 
is elementary, the standard comparison fluid is not. If we consider 
instead an object particle referred to a single comparison particle, both 
having exact momentum vectors, the answer is simple. The R.C. matrix 



11-9] The Riemann-Christoffel Tensor 209 

is the outer product of their stream vectors. Already in Chapter x, the 
portion of this R.C. matrix which specifies linkage of translations, viz. the 
kinematical energy tensor, has been called the "mutual energy tensor" 
of the two particles ; but this was merely a development of the nomenclature 
of current quantum theory. The connection with energy as classically, 
i.e. macroscopically, defined appears for the first time in the present Chapter. 

The standard comparison fluid has an isotropy in space and time which 
can be defined by the condition that its energy tensor (referred to the 
frame) is invariant for rotations and Lorentz transformations, or equi- 
valently by the condition that its energy tensorf is to be identified with 
the metrical tensor g^. To pass from an elementary comparison particle 
to this standard comparison fluid, it is not sufficient to integrate over 
a symmetrical distribution of stream vectors of the comparison particle. 
The symmetry is disturbed by the recoil due to the motion of the object 
particle (specified by its exact stream vector). To neglect this recoil would 
be to neglect the very thing we are investigating, namely the mechanical 
specification of the object particle ; for in the idealised universe consisting 
of an object particle and comparison fluid, the only manifestation of the 
mass of the object particle is in its mechanical reaction on the com- 
parison fluid. 

In treating the construction of the R.C. matrix, and the connection of 
mass and momentum in quantum theory with the corresponding macro- 
scopic tensors, it is essential to bear in mind that displacement of the 
object particle disturbs the comparison fluid, the combination of action 
and reaction being expressed by double wave functions. But in applying 
this theory to the elementary wave functions of quantum theory, the 
complication arises that these are adapted to a different point of view. 
These simple wave functions are intended to describe self-contained 
systems superposed on an undisturbed environment. The conception of 
such detached systems is essentially non-relativistic, but it can be justified 
up to a point as a practical procedure. It is evidently necessary that the 
energy and momentum associated with the simple wave functions should 
include, not only that which properly belongs to the object particle, 
but also the recoil energy and momentum communicated to the comparison 
fluid. We have therefore to separate out from the whole R.C. matrix the 
constant portion which corresponds to the undisturbed comparison fluid 
(and provides a constant metric) ; the remainder is then regarded as the 
self energy of entities which have been added to the comparison fluid 
without disturbing it. The actual calculation of this self energy, for an 
elementary particle added to the standard comparison fluid, will be treated 
in Chapter xn. 

| More precisely, its kinematical self energy tensor. 

ETP 14 



210 Physical Applications [1 1-0 

A much easier problem is that of an object particle with an entirely 
uncertain momentum vector, whose recoil accordingly does not disturb 
the symmetry of the comparison fluid. Complete uncertainty involves 
uncertainty of charge and spin, so that the particle is necessarily neutral. 
To calculate the corresponding E.G. matrix we project the uncertain 
stream vectors of the object particle and comparison fluid from the 
EF frame to the CD frame. In the latter frame the stream vectors 
become exact; we therefore form their outer product, and project back 
into the EF frame. The resulting R.C. matrix is of the form (11-721), and 
gives a R.C. tensor which represents de Sitter space-time. 

We can regard the standard comparison fluid as composed of N neutral 
particles with uncertain momentum vectors, and with exact position 
vectors uniformly distributed in space and time.f Generally in treating 
a single object particle we use only one of these particles as comparison 
fluid a partial comparison fluid. In any problem relating to the actual 
universe the N particles of the comparison fluid must be present as object 
particles, although only a few of them are mentioned explicitly; the 
remainder are either unspecified or specified macroscopically. It is there- 
fore convenient to pair the particles, and treat each object particle as 
disturbing only its allotted comparison particle. The substitution of a 
partial for a total comparison fluid magnifies the recoil, and therefore the 
apparent mass, in the ratio N] this factor is absorbed into the ratio of 
the units used respectively in macroscopic theory and quantum theory, 
the elementary equations and definitions of the latter theory being, as it 
were, based on the assumption that the particle under consideration is 
the only particle in the universe. We shall find later that more precisely 
the factor is fJV; the modification is due to the exclusion effect of the 
particles of the comparison fluid on one another. 

When all the particles are unspecified we obtain the cosmical R.C. 
matrix. Extracting from it the macroscopic R.C. tensor, we find that it 
corresponds to de Sitter metric. The energy tensor T^ v vanishes; this 
is the relative tensor of the aggregation treated as object system referred 
to the same aggregation treated as comparison fluid. The relative energy 
tensor is closely related to the mutual energy tensor (which in this case 
becomes a self energy tensor) ; but in the mutual or self energy tensor 
the kinematical and electrical components are kept distinct, and the 
former constitute the energy components in the more limited sense. The 
kinematical self-energy B^ does not vanish, and, as already explained, 
it is identified with g^ v . 

To obtain any other than a de Sitter metric it is necessary to specify 

f This is for analytical convenience; the actual distribution in position is, of course, 
continuous. 



1 1-9] The Riemann-Christoffel Tensor 21 1 

some particles macroscopically. The only important alternative metric is 
that of an Einstein universe ; this will be treated in Chapter xiv. Irregular 
metrics are important in general relativity theory, but it seems un- 
profitable to combine them with microscopic problems, e.g. to investigate 
the behaviour of a hydrogen atom in an irreducible gravitational field. 
Practically the perturbation would be exceedingly minute; and if we 
retain such small quantities, it is presumably illegitimate to employ a 
macroscopic specification of the perturbing system. In any case the 
problem is concerned with a special kind of perturbation of the atom and 
it would be out of place to deal with it in general theory. For the 
purposes of quantum mechanics we are therefore limited to the two 
possible uniform metrics, viz. the de Sitter and Einstein universes. 

The prominence of the de Sitter metric in our investigations is due to 
the fact that, following Dirac, we have developed the theory from Lorentz- 
invariant equations. Lorentz transformations are not applicable to an 
Einstein universe. So long as we treat free particles, with exact momentum 
vectors and therefore distributed uniformly throughout the universe, 
a Lorentz-invariant frame of de Sitter type is required. When we come 
to less abstract problems, and treat steady states, represented by internal 
wave functions to which Lorentz transformations are inapplicable, an 
Einstein metric becomes admissible; and it is to be preferred because 
(unlike the de Sitter metric) it forms a static reference system. Current 
quantum theory in its more abstract formulae presupposes a de Sitter 
background, and in its more practical formulae presupposes an Einstein 
background. Perhaps the most difficult part of the present investigation 
has been the sorting of these two influences. 



CHAPTER XII 
THE MASS-RATIO OF THE PROTON AND ELECTRON 

12* 1 . Contraction of a Volume Element. 

This chapter deals with a point which arises when the double strain vector, 

<S-Z7F-* B ^ Xr 8 , (12-111) 

of an elementary particle, with another elementary particle as comparison 
fluid, is contracted to form the simple strain vector, 



which represents the same particle with a neutral particle as comparison 
fluid (H-84).t 

We may remind ourselves how this process arises. From an observational 
point of view, it is meaningless to describe a particle without some physical 
reference object (11-1); and the observable characteristics which we 
measure are properties of the combined system of the particle and reference 
object. The wave functions containing observable characteristics, such as 
the mass of the particle, must therefore be double wave functions Y, X, of 
the particle and an idealised standard reference object which we call the 
comparison fluid. But in current quantum theory the mass and other 
characteristics of the particle are assumed to be contained in simple wave 
functions 0, x- How comes it that double wave functions Y, X can be 
replaced in practice by simple wave functions 0, #, apparently without 
much harm? The first part of the answer was reached in 11-8, when we 
found that, by adopting the probability distribution of a neutral particle as 
comparison fluid, the double wave vectors Y = a <0, X = Xy w S are con ~ 
tracted with respect to the suffixes j8 and 8 which refer to the comparison 
fluid; so that, for some purposes at least, we can ignore the second suffix 
and treat Y and X as simple wave vectors. In particular the double strain 
vector S is contracted to {/S}, and becomes the product (actual or symbolic) 
of two simple wave vectors ifi, #, together with a merely algebraic factor 
<f>pa>p. 

It is not our business to supply a complete defence of this substitution of 
simple wave vectors for double wave vectors. It is a substitution which 
current theory has inadvertently made; and we have to trace how its con- 
sequences appear in the formulae of current theory. For our own part we 
shall be continually going back to the double wave vectors to see what 
current theory has missed by this substitution. 

t The factorisation in (12-111) and (12-112) need only be symbolic. 



12-1] The Mass-ratio of the Proton and Electron 213 

The strain vector (12-111) specifies an elementary configuration of the 
combined system represented by a single point in its phase space. The 
actual state is specified by probability factors attached to a number of such 
configurations, or more generally by a continuous probability distribution 
over the phase space. In the latter case S is normalised so that Sda)jfl is 
(apart from a unitary factor containing the phases) the probability of a 
range of configurations dw ( 7-7). 

The question that we have to consider is, What happens to dw when we 
contract (12-111)? 

Our procedure is based on the law of multiplication of probabilities. The 
intention is that, leaving out the unitary phase factors, S=UV shall 
express the law that the probability of a configuration of the double system 
described by 8 is the product of the probabilities of the corresponding 
configurations of the simple systems described by C7 and V. But SUV 
does not express this unless volume elements are inserted on both sides. 
The formulae as they stand postulate that the different configurations 
are represented by discrete wave functions; they can only be extended to 
continuous wave functions if it is assumed (or arranged) that the volume 
elements take care of themselves. 

It will now be convenient to change the notation. Let /S, $', 8 Q be the 
strain vectors of the object system, comparison fluid and combined systemjf 
and let dw, da>', da> Q be volume elements in their respective phase spaces. 
To neutralise the comparison fluid we add together four elements da> Q which 
yield the required balance of charge and spin; and we equate the probability 
contained in these four elements to the product of the probabilities in the 
corresponding elements of the phase spaces of 8 and S'. Denoting the 
summation of four elements by { }, this gives (disregarding unitary factors) 

{S }dW"o= (Sda>l&) (S'dw'IV). (12-12) 

In order that this may reduce to {$ } = 88' as currently assumed, we must 
have dw Q =kda>dw' 9 (12-13) 

where k is the constant i2 /i22'. 

The point at which the assumption {$ } = 88' enters fundamentally into 
current theory is in the dynamical equations (8-4). As explained in 9-1, 
the significance of these is that they give "steady states" distinguished by 
the constancy of some observable characteristic. Primarily therefore the 
dynamical equations apply to the double space vectors T of a particle and 
comparison fluid, since the particle alone has no observable characteristics. 
But in practice we substitute the simple space vectors T, T' which express 
(unobservable) relations of the two systems to a geometrical frame. Whether 
these will satisfy the equations, i.e. whether T and T' will be parallelly 

f For brevity, the system whose strain vector is 8 will be called "the system S". 



214 Physical Applications 

displaced when the double vector T containing the recognisable character- 
istic is parallelly displaced, depends on their relation to T Q . If T = TT' 9 
parallel displacement of T and T r involves parallel displacement of T Q . 
But this is not in general true if T do> = TT'da>da>', since volume elements 
are not transformed by parallel displacement in the same way as space 
vectors. The dynamical equations currently adopted therefore impose an 
additional condition TT' = T , or in the present application TT' = {T }. The 
corresponding strain vectors therefore satisfy SS' = {S }. 

Equation (12- 13) results from a comparison of two conditions, 

{S Q }da> = kSdto . S'da>', {S Q } = SS'. (12-14) 

The first is imposed by the law of multiplication of probabilities, and the 
second by the dynamical equations. 

We can now define more narrowly the circumstances in which (12*13) has 
to be satisfied. Since it represents an assumption in the dynamical equations, 
it applies to displacement in the dynamical coordinates. Further, since the 
dynamical equations are differential equations of the second order (9-2), 
we have to retain squares of the coordinates in the investigation. Local 
orthogonal coordinates in phase space are insufficient; and it will be neces- 
sary to use a more extended system such as stereographic coordinates. 

Before applying (12-13) we must set forth certain general considerations 
which arise in combining two systems. For purposes of exposition it is more 
convenient to consider two elementary particles S 9 S' ; but S' will ultimately 
be replaced by a neutral particle. 

12-2. Combination of Two Systems. 

As an approach to the more general theory, let us first consider the com- 
bination of two particles S, S' into a single system S according to the 
elementary wave mechanics of de Broglie and Schrodinger, in which each 
particle has only one phase variable. Let 0, 0' be the phase variables, so that 
the wave functions of S and S' are 

<A = e**V , 0' = e*^ '. (12-211) 

These are combined by multiplication, so that, setting *F = ^r ^r ', the wave 
function of the double system is 

T = e**< '+*'>%. (12-212) 
As in 7-4, we introduce linear variables by setting 

ds = Rde, d8 f = R'd8'. (12-213) 

Then, writing m 1/21?, m' = 1/25', (12-214) 

we have T = e******^ . (12-215) 

In this form the phase angle ms + m's' can be interpreted as action, the 
action of a system being Jracfo. 



12-2] The Mass-ratio of the Proton and Electron 215 

We have described Y as the wave function of a double system. That is not 
quite the same thing as the wave function of a combined system. To obtain 
the wave function of the combined system, the time coordinates t, t' of the 
two particles must be equated. Consider, for example, the sun-earth system. 
That does not comprise a combination of the earth today with the sun a 
week ago; no reference to the "orbit" of such a combination will be found in 
astronomical textbooks. The essence of the process of "combining" is the 
substitution of a single time coordinate for the whole system instead of 
independent time coordinates for its separate parts ( 7-8). 

We are not arguing that there would be anything illegitimate in the con- 
ception of 8 and 8' as a double system, i.e. with two independent time 
variables. Our point is that there are two possible conceptions whicli must 
be carefully distinguished, and that the usual description of a composite 
system refers to the conception of it as a combined system, i.e. with a single 
time variable. When we speak of the orbit of a planet or the quantum state 
of an atom we are referring to the combined system, not the double system. 

When s and s' are expressed as functions of the coordinates, the wave 
function T of the double system is a function of eight coordinates x, y, z 9 1, 
a?', y', z', t'. The wave function of the combined system S Q is (*F) r _,, a function 
of seven variables. The mass or energy of S is the value of the operator 
-id fit, which by (12-215) is 

mdsldt + m'(d8'ldt') t , H . 

This is the sum of the masses of S and S' at a simultaneous instant, the 
factors ds/dt, ds'/dt' being the FitzGerald factors representing change of mass 
with velocity. 

In the general theory the simple algebraic phases 0, 0' are replaced by 
space-like matrices with ten components (7-3) which, for a range of con- 
figurations small enough to be referred to local orthogonal coordinates, may 

be denoted by = SJEf0=SJ5?a;/-R, 1 

s n n^ t p p/ I (12-221) 

and the wave functions (within the above small range) are 

0=e**0 , 0' = e**'0 '. (12-222) 

For the wave function representing a combined system S Q we have similarly 

T=c**T , (12-231) 

where t^^^E^O^^^E^x^B^. (12-232) 

In forming a combined system out of S and S' we equate their times t and 
t', and this common time is also the time t of the combined system S . 
Since the time in phase space is represented by the algebraic coordinate, the 
identification t = t' = J becomes, in the above notation, 

(12-24) 



216 Physical Applications [12-2 

We notice that, whereas in Schrodinger's theory the algebraic phase 
variable represents s, in the general theory it represents t. The difference is 
not surprising, since in the former theory one variable has to do the work of 
ten. In the matrix theory the strains of the system corresponding to dis- 
placements along x, y, z are provided for more precisely by separate non- 
algebraic phase variables, and therefore no longer enter into the algebraic 
phase. A possible misunderstanding may be caused by the fact that in matrix 
theory just as in Schrodinger's theory the wave function for plane waves is 
i// = e (m8 *// Q ; but here ims is the eigenvalue of a matrix expression representing 
the phases (i/r having been specially chosen as an eigensymbol), and is not 
to be identified with the algebraic phase Ji0 16 . 

It may be well to state again the reason why the time is represented by 
the algebraic coordinate in phase space. Progress in time corresponds to 
rotation (about the centre of curvature of space-time) in the plane associated 
with jB 45 . Hence energy conjugate to the time is the 45 component of the 
complete space vector which comprises the mechanical properties of the 
particle. The space vector is multiplied by iE^ to form the associated strain 
vector; energy is therefore the E IB component of the strain vector. The trans- 
formation g = ei fcl iA applied to the strain vectors in phase space, which 
causes displacement in the algebraic coordinate, is therefore a displacement 
conjugate to the energy, i.e. displacement in time. 

The phase space of $ contains 136 coordinates v . It may be suggested 
that since the configurations of S and S' are each completely specified by 
ten coordinates, one of which they have in common, we shall require a phase 
space of 19 dimensions at most to specify the combined system $ . But that 
is to begin at the wrong end of the problem. Our observational data relate 
to complex systems. We do not construct S Q to represent data which have 
been ascertained about S and S'', we construct 8 and S' to represent data 
which have been ascertained about $ . We have to take a system repre- 
sented by a double wave vector in any combination of its 136 phases, all 
combinations being relativistically equivalent, and determine the conditions 
under which it can be dissected into two systems each with ten phases, 
or into a ten-phase system and a neutral comparison fluid with a single 
phase. 

Initially the probability of $ is distributed uniformly over its 136- 
dimensional phase space. We replace it by two initial probability distribu- 
tions uniform over the ten-dimensional phase spaces of S and S'. A great 
variety of configurations regarded as potentially distinguishable in the first 
representation must be classed as indistinguishable in the second representa- 
tion. That is to say, there is an alteration of the basis of statistics. It will be 
found that this alteration has important consequences. 



12-3] The Mass-ratio of the Proton and Electron 217 

12*3. The Augmented Phase Space. 

In combining 8 and S' we drop a time coordinate. Accordingly before 
dissecting $ we must insert an additional time coordinate and treat S as 
having an infinitesimal uniform extension in this extra time. Let 



so that dt Q dt f =dtdt'. (12-32) 

Reducing to angular measure with the respective scale constants of the 
three phase spaces, (12*32) becomes 

R Q *dO Q dO r =RR'dO lB dd lB '. (12-33) 

By this augmentation the phase space of S Q contains two algebraic 
phases ( = 16 16 ) and r . Owing to their commutative property we can 
treat the probability distribution in the algebraic phases independently of 
the distribution in the other phases (cf. 7-7). For example, in ten-dimen- 
sional phase space the volume element in local orthogonal coordinates 
daj = d0 1 d0 2 . . . contains an amount of probability do>/2, where Q is the whole 
volume of phase space which we have shown to be finite ( 7-4). Writing daj c 
and l c for the corresponding volumes without the algebraic dimension, we 

haVe dco = d0 16 do> c , Q = 27rii c , (12-34) 

and the probability dw/Q is the product of the independent probabilities 
d0 16 /27T and do> e /!i c that 16 is in the range d0 16 and that the other coordinates 
are in the range da) c . 

Hence the initial probability associated with the range d0 16 d0 16 ' is 
(d0 16 /27r) (dO lB '/2Tr) and the initial probability associated with the range 
d0 d0 r is (d0 /27r) (dO r /27r). These must be equal, since rf0 16 d0 16 ' and d0 d0 r 
represent the same range described in different ways. Hence by (12-33) 

R*=RR'. (12-35) 

The new variable r must, like the other phases, correspond to a circular 
(not hyperbolic) transformation, since it is essential that phase space shall 
be closed (7-3). We cannot begin the process of dissection of S unless this 
condition is satisfied. It is, however, desirable to examine an argument 
which seems to suggest that t r should be represented by a hyperbolic 
transformation. 

The definition r = r /JR refers to local orthogonal coordinates, and for 
extended coordinate systems the relation of t r and r becomes non-linear. 
Adopting stereographic coordinates and considering variations of t r only, 
we have, by (7-58), 



so that r = 2tan- 1 ( r /2JB ). Thus, as r increases, t r increases without limit, 



218 Physical Applications [12-3 

provided that O r is real. If, however, r is an imaginary quantity iu r + we 
replace t r by it r and obtain 



so that w r ==2tanh~ 1 (^/2JZ ). Then as w r ->oo, r ->2JR . Accordingly -' is 
not greater than 2R . It suggests itself that 2R should be identified with the 
distance between S and 8', and therefore with the light-time (the velocity of 
light being unity). The result t t' > 2R then means that two systems cannot 
be combined at instants such that one is in the absolute future of the other. 
The variable u r is found to be important in another problem which arises 
in the combination of two particles ( 15-5), but here it is irrelevant. The 
radius R of phase space is a scale constant applicable to all configurations 
represented in phase space. If we equate 2JR to the distance between the 
particles, the phase space can contain only those configurations for which the 
distance is a fixed constant. That is not the problem here treated. 

12-4. The Fundamental Quadratic Equation. 

Writing da) = V g.dr as in (7*422), the relation (12-13) becomes 

(12-41) 



The formula now applies to any system of coordinates. 

We adopt stereographic coordinates. It has been shown in 7-5 that when 
a transformation (not necessarily infinitesimal, but not involving anti- 
perpendicular components) is applied, each point of phase space in the 
infinitesimal neighbourhood of the origin receives the same increment of its 
stereographic coordinates. That is to say, when a volume element is dis- 
placed in phase space dr remains constant. Hence for such displacements 
in the three phase spaces we must have 

V^ = CV^.V^V, (12-42) 

where C is a constant. By (7-59) 



where r 2 is the square of the length of the displacement, and n is the number 
of dimensions of the phase space. Inserting this in (12*42), we have 

(1 + r 2 /4# 2 )-o = (i + r 2 /4# 2 )-* (1 + r' 2 ^' 2 )-"' (12-43) 
together with JB - W = CR" n R'~ n '. 

Consider a displacement in time, so that r, r', r are # 16 , # 16 ', # 16>16 , and 
are all equal by (12*24). Then, expanding (12*43) and equating coefficients 
of r 2 , we have n /Itf=n/IP+n'/Il'*. (12*44) 

By (12*35) and (12*214), 1Z 2 = BR', and R/R' = m'/m. Hence 

'*=0. (12-45) 



12-5] The Mass-ratio of the Proton and Electron 219 

For an elementary particle and a neutral comparison fluid, the dimensions 

of the phase spaces are 

'!. (12-46) 



(The strain vector of a neutral particle, being algebraic, has only one phase 
variable.) Hence we obtain the fundamental quadratic equation 

10w 2 -136ww' + m' 2 = 0. (12-47) 

Its two solutions give two possible masses m for an elementary charged 
particle in terms of the mass m' of a neutral particle. These accordingly will 
be the masses of the proton and electron. The ratio of the two roots is 1847-6. 
This is the ratio of the masses of the proton and electron, when mass or 
energy is defined as in quantum theory by the operator ( - iA/2?r) 3/3*, or 
equivalently by E = hv. We shall find in 15-9, that this deviates somewhat 
from the classical definition of mass and energy; and that, adopting the 
classical definition, the theoretical mass-ratio is 1834-1. Since all our 
formulae imply the quantum definition, we must adhere to the value 
1847-6 in the developments which follow. 

12*5. Notes on the Solution. 

One or two steps in the foregoing proof require fuller discussion. 

Should we take M O = 136 (original phase space) or 137 (augmented phase 
space)? Equation (12-42) was obtained by considering parallel displace- 
ment of a volume element dr conformably with the dynamical equations. 
The angular element dO r is not a range dynamically transferred from the 
origin by parallel displacement; it is a constant infinitesimal thickness 
assigned to phase space, and hence equal for all volume elements considered. 
The value w = 136 is therefore correct. 

Is it legitimate to satisfy (12-43) as far as r 2 , leaving a discrepancy in the 
higher powers of r? Equation (12-43) is not valid beyond r 2 ; its exact form 



1 ', (12-51) 

where J , t, t' are corresponding times in the three systems measured in 
stereographic coordinates. The times are equal in some system of reckoning, 
but there is no reason to suppose that that reckoning is stereographic. In 
fact for large values of t the stereographic reckoning becomes absurd. The 
substitution t t t' =r is only valid to the first order of small quantities, 
and therefore (12-43) is only valid to the second order. To the first order 
(corresponding to natural coordinates) the reckoning of t is fixed by the 
scale constant R, and there is no ambiguity as to what is meant by the 
three systems being simultaneous. In short, having found m/m' from the 
terms in t 2 , the higher powers merely tell us how we must define simultaneity 
of the three systems for large values of t. 



220 Physical Applications [12-5 

By obtaining a formula correct as far as r 2 , we obtain exact values of the 
second derivatives at the origin; so that we achieve our aim of providing for 
the second order differential equations which express the laws of physics. 
Results obtained in a special coordinate system, such as stereographic 
coordinates, are of no great interest in themselves. We use them to extract 
the invariants, in particular the curvature invariants which correspond to 
mass and density. For this purpose it is only necessary to expand as far as r 2 . 

We come to a more difficult point. The displacement which we have con- 
sidered is along the algebraic coordinate; but the algebraic coordinate is 
very much aloof from the others; and although we have used the same stereo- 
graphic projection for it as for the others, there is no obvious need to do 
so. It is not possible in an extended region to use natural measure for all 
coordinates; but it is possible to use natural measure for the algebraic co- 
ordinate and stereographic measure for the rest. Geometrically the distinc- 
tion is that the algebraic dimension has cylindrical curvature, whereas the 
others by their non-commutative relations determine a spherical curvature. 
Spherical curvature is an invariant; and on the assumption that E, R, R Q 
are radii of spherical curvature, the result (12-44) is independent of the 
special system of coordinates used in obtaining it. Unfortunately, in the 
equation as derived, they are radii of cylindrical curvature. To validate the 
proof we must show that there is some condition (not yet mentioned) which 
requires the algebraic coordinate to be represented uniformly with the 
others. 

The fact is that we have not explicitly introduced the condition that the 
particle whose mass we are seeking is a pure elementary particle, having 
therefore a singular strain vector. As shown in 7-9 such a particle will have, 
associated with it, a singular line in phase space given by d 8 = ( E l + J57 16 ) d<f>. 
This provides an absolute scale comparison of lengths in the A\ and the E 16 
directions just as in four-dimensional space-time a singular line (con- 
stituting a light track) provides an absolute scale comparison between 
intervals of space and time. 

To go back to the first principles of measurement magnitudes in different 
directions are compared on the basis that "equivalent" magnitudes (which 
can be transformed into one another by relativity rotations) are equal 
magnitudes. This enables us to compare magnitudes in perpendicular 
directions immediately, and magnitudes in antiperpendicular directions 
indirectly, using for the latter a direction perpendicular to both as inter- 
mediary. But this does not apply to the algebraic direction which has no 
direction perpendicular to it. The measurement of proper mass, proper 
energy, proper time, in terms of the standards used for measuring momenta 
and lengths in other directions, must depend on a different principle of com- 
parison. This is supplied by the formula (6-64) for the strain vector of a pure 



12-6] The Mass-ratio of the Proton and Electron 221 

particle. Its momenta in four antiperpendicular directions E l9 /? 23 , J5? 45 , j 16 
are equal. This enables us to compare the momenta in the E 1Q direction with 
the momenta in other directions. If, as suggested above, a different scale 
constant J? were employed for the J? 16 direction, a stream vector which is 
pure in angular measure would not be pure hi linear measure; and our whole 
treatment of the analysis of matter into pure wave functions representing 
elementary particles would have to be revised. 

We may put the result in another way. An elementary particle possesses 
spin. We can measure time by the spin coordinate of the particle as time 
is in fact measured by the spin coordinate of the earth. Thus x l9 x n can be 
substituted for # 16 , #i 6 ,i 6 in our previous deduction; and the difficulties 
arising from the peculiar character of the algebraic coordinate are then 
evaded.f This brings out the fact that the mass which we have determined 
is that of an elementary particle. A distinctive feature of the elementary 
particle is that it has four dynamical coordinates all of which measure the 
time; but two of them are not shown in phase space. 

12-6. Energy Invariants. 

The primary operation of wave mechanics is multiplication; and in the 
combination of two systems the product of the masses m^m^ is more funda- 
mental than the sum m^m^. This was illustrated in the transformation to 
relative coordinates, in which the original particles are replaced by external 
w i internal particles whose masses Ji/, p, satisfy M/Z = m 1 m 2 . 

We shall call m^m^ the mutual pressure invariant of the two particles. 
Correspondingly m x 2 is called a self-pressure invariant. 

Let us provisionally interpret the term " pressure " literally. In statistical 
mechanics the pressure is the energy associated with one degree of freedom. J 
If there is equipartition between n degrees of freedom, the energy is nm l m 2 . 
We shall call nmm 2 the mutual energy invariant. The self-energy invariant is 
defined similarly. 

The fundamental equation (12-45) 

n^mm' = nm 2 + n'm' 2 

expresses that the mutual energy invariant of the system $ is the sum of 
the self-energy invariants of the two parts composing it. We thus get a kind 
of physical picture of the significance of the fundamental quadratic. In 
particular the equation I 36mmo== io m 2 + m a (12-61) 

t The algebraic coordinate was employed because we knew what were fct corresponding" 
displacements of the three systems in that coordinate but not in any other coordinate. 
But the above argument shows that, besides (12-24), we have x l = x ll . 

J We have in mind waves in which the energy is half kinetic and half potential. If kinetic 
energy alone is present a factor is introduced. 

We shall in future denote the mass of the neutral comparison particle by m instead of 
m' since m' seems a rather unsuitable notation for an important constant of nature. 



222 Physical Applications [12-6 

means that the introduction of the elementary particle ra can be regarded 
either as replacing a neutral unspecified particle m by a double system with 
136-dimensional phase space or as adding a simple system with ten-dimen- 
sional phase space. Both interpretations give the same total energy in- 
variant. From this point of view the number of dimensions of phase space 
is involved, because in the steady state represented by the initial probability 
distribution, each dimension shares in the equipartition of energy. 

Let us extend this to a static system formed by two elementary particles 
with a neutral comparison fluid. We know that two protons or two electrons 
cannot form a static system; and we may therefore anticipate that the two 
particles will be a proton and electron. In short, our system is a hydrogen 
atom (or possibly a neutron). 

We resolve the hydrogen atom as in 10-9 into an external and internal 
particle, described by external and internal wave functions, with masses 
M = m p + m e , p. = m p m e /(m p + m e ). (12-62) 

Since m p , m e are the roots of 10m 2 - 136wm + w 2 = 0, we have 

A/ = (136/10) w , ^=(1/136),,. 
Writing these in the form 

136jiw =10*f 2 , 136^wi = m 2 , (12-63) 

we see that for an external particle the mutual energy invariant of the 
particle and its comparison fluid is equal to the self-energy invariant of the 
particle; but for the internal particle the mutual energy invariant is eq 
to the self-energy invariant of the comparison fluid. The two particles have 
therefore a very different type of relationship to the comparison fluid. 

We can extend the foregoing definition of pressure and energy invariants 
to conditions in which the energy is represented by a symbolic operator W. 
The self-energy invariant of a particle in phase space of n dimensions is 
then n W 2 . If the wave functions representing steady states satisfy Dirac's 
equation W$ = m[i 9 it follows that (JP/ra)^ = ra0. Thus we have two oper- 
ational forms of the energy w, namely the linear hamiltonian W and the 
second order hamiltonian W 2 /m. For the external particle 

(external particle), (12-641) 

where y e is the natural constant 136m , and n is the number of dimensions 
of the phase space of the external particle. For the internal particle 

136F 2 (internal particle), (12-642) 



where y i = y e /136 2 

and n is unity. Thus y i9 y e are the factors which convert energy invariants 
into energies on the ordinary scale. 



12-7] The Mass-ratio of the Proton and Electron 223 

For an electron or proton, the fundamental quadratic can be written as 



so that the operational form of the energy is 

- (nW 2 + 2 ) (elementary particle). (12-643) 

7e 

It is interesting to compare (12-643) and (12-641). 

I think that for theoretical purposes the energy invariant is to be pre- 
ferred to the energy; that is to say, we should replace the quadratic energy 
operator JF 2 /w by nW 2 /y. In the following chapters I have not used the 
energy invariants as much as I might have done, fearing that it would be 
a stumbling-block to the reader. When the form W 2 /m is used, it should 
be realised that m is a combination of the universal constant ra with a 
geometrical factor of the problem, the latter introducing the number of 
degrees of freedom. It is also particularly important to notice that m 
is not necessarily the rest energy; for internal wave functions (which 
are those most concerned in practical problems) the rest energy is zero 
(10-982). 

12*7. The Association of Mass and Charge. 

We have found that an elementary particle must have one of two masses 
m p , m e . We shall now show that in static conditions the particles of mass 
m p are of one sign (arbitrarily called positive), and the particles of mass 
m e are of the other sign; also that the positive and negative charges are 
equal. 

To test the sign of the charge we introduce a field of uniform electrostatic 
potential * 4 . If we can show that the effect of the field is to make the mass 
m* of an elementary particle at rest satisfy the modified equation 



0, (12-71) 

where a is a function of /c 4 , the required conclusion follows. For then 

m p * + m e *=^m Q = m p + m e , (12-72) 

so that we may write m p * = m p + c, m e * = m e , (12-73) 

showing that the part of the mass or energy due to the existence of the field 
is equal and opposite for the two types of particle. Our problem therefore 
reduces to showing that * 4 affects only the last term of the quadratic. 

By 8-8 the constant potential * 4 can be removed by applying the gauge 
transformation 0' = e ilf **4^ to wave vectors of index 1. The strain vector 
of index 2 generating phase space then undergoes the transformation 



224 Physical Applications [12-7 

g> = e 2tK.x 4 s = c -2ic 4 / # g a i r eady contains the time factor e* e * = e***; so that 
the time factor of S' is e ii>IR , where 



4 lZ) = (l-2ic JK). (12-74) 

Thus the field-disturbed environment of the object system is converted 
into the standard environment (neutral space-time) by antedating our 
description of it.f In analysing ( 12-4) the double wave functions, which 
contain the observable relations, into a simple wave function of the particle 
and a one-phase wave function of the comparison fluid, we gave to all three 
functions the same time t\ if, however, we give the one-phase wave function 
a time t', it will equally represent a neutral comparison fluid at time t' or 
the actual field-disturbed environment at time t. With this modification 
(12-43) becomes 

(1 + * 2 /4JR 2 )- w = (1 + * 2 /4jR 2 )-' 1 (1 +f i/4/Z' 1 )-*' (12-75) 

and gives the condition for analysis of the double wave functions, containing 
the observable relations of the particle to its field-disturbed environment 
at time t, into a simple wave function of the particle and a one-phase wave 
function of its field-disturbed environment at the same time t. The change 
only affects the last term of the quadratic equation, which becomes 

nra 2 -n ram' + (e7$) 2 m' 2 = 0. (12-76) 

This is the result we required. 

We do not apply the gauge transformation to the double wave function, 
because that contains the observable relations, and it is understood that 
the energy of the particle is to be determined from observation precisely as 
if the actual environment were the standard environment. As we should 
ordinarily say, the change from m to m* is an apparent change of mass, 
due to the difference between the comparison fluid actually present and 
available for our observations and the ideal comparison fluid referred to in 
the definition of m. 

The value of a in (12-71) is (l-2/c OJ R'); or, since ll'2R f = m 9 (formerly 
Call6dm/ >' a=l-K /m . (12-771) 

Using (12-73), we derive the relation 

-AWiM 2 . (12 . 772) 

>e) \ / ' 

Thus a is real for a physically real electrostatic potential, and (for electro- 
magnetic fields which are not too extravagantly great) is also real. By a 
happy accident, positive electric potential corresponds to positive * . 
The numerical value of a does not much concern us; but if any application 

t The reader may perhaps be inclined to object that, if (as usual) the comparison fluid 
is at rest, antedating makes no difference. But there is a phase angle which changes with 
the time; and we see above that it is this phase angle which exhibits the antedating. 



12-7] The Mass-ratio of the Proton and Electron 225 

is made of (12-771), we should notice that /c is not there measured in the 
same units as in the ordinary wave equation since it refers to a gauge 
transformation of the comparison fluid. 

It is interesting to consider the physical basis of the dissymmetry of 
positive and negative charges. Formally they are correlated to the two 
square roots of 1. This can only give rise to observable dissymmetry (such 
as a difference of mass) if, elsewhere in the theory, the two square roots have 
been allotted unsymmetrical roles in the description of phenomena. In the 
expression for a physically real interval E l x i + E z x 2 -{- E^x^ + E^it we have 
to select one of the square roots of 1, which we shall denote by iy, such 
that Etijt represents displacement towards the future when t is positive. 
Then any other square root of 1 occurring in our formulae is distinguished 
as if or ij . 

The irreversibility of time is manifested in three ways: (a) in consciousness, 
(6) in the laws of entropy, (c) in the cosmic expansion. Consciousness must 
be regarded as the ultimate source of the irreversibility, at any rate from the 
point of view of physical theory. Owing to the curious fact that our minds 
are acquainted by sensory mechanism with the past but not the future, 
observational knowledge has the form of an integral over past time up to 
the present moment t. Since probability is relative to knowledge, a formula- 
tion of the universe in terms of probability distributions is exceedingly 
unsymmetrical with respect to past and future time; and this irreversibility 
is shown in the manifestations (b) and (c). It is also shown in wave mechanics 
in a more elementary way by the concentrated wave packets, which are 
formed discontinuously by our observations and diffuse continuously as t 
increases towards the future. 

For the distinction of protons and electrons the most relevant mani- 
festation is the cosmic expansion. I do not mean the actual "expansion of 
the universe", which might perhaps have been reversed by altering the 
initial conditions. But a region of space with de Sitter metric necessarily 
expands relatively to our standards of measurement. We generally treat a 
small region of space isolated from the rest by artificial boundary conditions; 
but the dilemma, first pointed out by de Sitter, always appears either we 
must take it to be a portion of static spherical space (an Einstein universe), 
in which case Lorentz transformations are inapplicable; or we must take 
it to be a portion of de Sitter space-time, in which case geodesies that are 
initially parallel proceed to diverge. Since we have introduced space-time 
by Lorentz transformations, our method follows the latter alternative. 
Correspondingly we have an expanding comparison fluid, which is un- 
symmetrically related to past and future time.- In particular, elementary 
particles symmetrically related to the expanding comparison fluid are not 
symmetrically related to past and future time. 



226 Physical Applications [12-8 

12-8. The Stern-Gerlach Effect. 

We shall now calculate the energy of a hydrogen atom in an external 
macroscopic electromagnetic field. Representing the atom by external and 
internal wave functions, we have to calculate the effect of the field on these 
separately. An essential point in the investigation is that the external wave 
functions are continuous and the internal wave functions are discrete. 

We use the method of gauge transformations explained in 8-8. Let us 
introduce a field of electromagnetic potential by changing the local unit of 
length, so that the scale constants R, R r of the phase spaces of a particle 
and its comparison fluid become in the new measure j8~ 1 JJ, /J- 1 ./?', where j8 
is a unitary complex factor. The coefficient V g in the reduction from co- 
ordinate volume to natural volume will be changed to ]8 10 V g for the ten- 
dimensional phase space and j8V gr for the one-dimensional phase space. 
We do not recognise complex measures of volume; accordingly the factors 
are transferred to the strain vectors which become /J 10 $, pS'. The transfer 
means that the change of probability distribution due to the electro- 
magnetic field is incorporated in the modifying factor instead of in the 
initial probability distribution. 

The masses of the particles are such as to validate the usual assumption 
that the volume elements can be ignored in the dynamical equations; that 
is to say, 8 and S' can be treated as discrete strain vectors in the dynamical 
equations. But they do not behave as discrete wave vectors for gauge 
transformations, and consequently the electromagnetic terms in the 
dynamical equations are not the same. Let us now consider the gauge 
transformation for a discrete strain vector. 

A discrete wave function exists only for discrete values of its parameters; 
but it is a continuous function of coordinates, so that it is affected by the 
gauge transformation of the volume element of coordinates. By the theory 
of Chapter vm a discrete wave function has four dynamical coordinates; 
whence it follows that the transformation of the strain vector is S->/3*S. 
We obtain the same result more directly from the ordinary expression 

W+V^gdXtdxidXi (12-81) 

for the probability in an element dx : dx 2 dx 3 of three-dimensional space. 
Here V g is the four-dimensional factor, defined in ordinary tensor 
calculus, which varies as ]8 4 .t The factor ]8 4 is therefore attached to the 
discrete strain vector function ^*. 

We have therefore the following result. Although the external wave 

function is continuous, it is permissible to treat it as discrete for the ordinary 

purposes of wave mechanics, the mass M assigned to it having been expressly 

chosen so as to validate this practice. But if a gauge transformation is 

f Mathematical Theory of Relativity, equation (85-44). 



12-8] The Mass-ratio of the Proton and Electron 227 

applied, the strain vector will be changed in the ratio J3 10 whereas the 
change for a discrete wave function would have been in the ratio j8 4 . The 
wave functions $, <f> are therefore changed in the ratio j3 6 , as compared with j3 2 . 
Applying the theory of 8-8 with A = j8 2 , j8 5 , if the potentials in the wave 
equation for a discrete wave function are K^, those in the equation for the 
external wave function are f/c^. Thus for the same electromagnetic field 

we have g 

p = - i - + ic (discrete wave functions), (12-821) 

^'li 

JP = i h f K (continuous wave functions). (12-822) 
o*i> 

The hamiltonian of the external particle is therefore 

-O < = S*' (12-831) 



i ^ 
which gives the second order hamiltonian 



If the field consists of a magnetic force eH = 3/c 2 /3# 3 - 3K 3 /3# 2 * n the ^ 
direction, the last term becomes (f ie/^/3/) E^ . To the classical approxima- 
tion this term is twice the mutual energy of the particle and electromagnetic 
field. The mutual energy is thus 

* ^-l.jiJS^. (12-84) 

The factor UE 23 is the spin momentum of the external particle in the plane 
normal to H and has eigenvalues J. The factor f e/M, or in the usual units 
|/MC, is the effective magnetic moment per unit angular momentum. 
The corresponding result for the internal particle is 

-^.Z, (12-85) 

where the operator Z includes, besides the spin momentum, half the orbital 
momentum. f The factor f is now omitted because the internal wave func- 
tions are discrete. But it is necessary to consider whether they are the 
standard discrete wave functions in four dimensions referred to in (12-821), 
or whether the fact that internal space is three-dimensional will not give 
K ' = f * , and an energy o A JJ 

.~-Z. (12-86) 

I think that for the theory of the atom, so far as it is developed in Chapter 
ix, (12-86) is the correct result, the factor f being essentially the same as 
that which will occur later in (14-153). It is true that the internal wave 
function has the standard number of dynamical coordinates, and appears 

f The factor & is the well-known magnetic anomaly. 

15-2 



228 Physical Applications [12-8 

in 9-2, 9-3 to be a function of t as well as of the space coordinates. But 
although t is used in Chapter ix in order to agree with the usual notation, t is 
not the time; it will be shown in Chapter xv that it is an interchange co- 
ordinate. Now the interchange coordinate is by its nature gauge-invariant. 
Thus for internal space, V g varies as j3 3 instead of j8 4 , and the factor f is 
required. 

But it is well known that, when a magnetic field is applied to an atom, we 
have to distinguish between weak and strong fields. A strong field cannot be 
treated as a perturbation of the normal states, but involves a re-analysis into 
elementary states. The particle is, as it were, torn in its allegiance between 
the planes of simultaneity determined by the external momentum vector 
and those determined by H^ (with time direction so chosen that the field is 
purely magnetic). This effect of strong fields is generally described as an 
"uncoupling" of spins. The magnetically determined planes of simultaneity 
introduce a genuine time coordinate into the internal state additional to the 
proper time s correlated to the external momentum vector; the original t 
(interchange coordinate) is then relegated to the role of argument for small 
perturbations, and the states are re-analysed with respect to the genuine 
time coordinate. It appears therefore that (12-85) is right for strong fields; 
but it will be replaced by (12-86) in weak fields, where a gauge-invariant 
interchange coordinate takes the place of the genuine time coordinate. 

It is to be understood that in our equations H is not necessarily a measure 
of the field in absolute units; it is used to connect (12*84) alternatively (but 
not simultaneously) with (12-85) or (12-86). Actually in passing from (12-85) 
to (12-86) we should change the unit of energy in proportion to the number 
of degrees of freedom, so that they represent the same absolute energy. 

By suitable arrangements a stream of particles, projected in a strong 
inhomogeneous magnetic field, can be made to divide itself according to the 
different combinations of eigenvalues of (12-84) and (12-85). From the 
measured deflections the ratio of the factors fe/Jif c and e/fjLC can be determined. 
The generally accepted results give the numerical coefficient f , agreeing 
with our theory. Since M and //, are very nearly equal to m p and m e , the 
magnetic energies are generally attributed to the proton and electron, 
respectively. But it is clear from the foregoing investigation that they 
properly belong to the external and internal motions, respectively. 

The result for weak fields also appears to be confirmed by experiment 
(Babi, Kellogg and Zacharias, Physical Review, 46, 157 (1934)). In this case 
the ratio is fe/jfc : f e//uc, so that the numerical coefficient is \-. The 
experimental result is stated to be 3-25 0-3. 

Whilst the result for strong fields is a simple consequence of the theory, 
the theory for weak fields may perhaps require a closer scrutiny than I have 
been able to give. 



CHAPTER XIII 
STANDING WAVES 

13-1. Scalar Wave Functions . 

Vector wave functions have superseded the scalar wave functions of 
Schrodinger in the problems of atomic physics; but there are certain applica- 
tions of wave mechanics in which scalar wave functions are still needed. 
These will now occupy us for two chapters. We shall develop the theory of 
scalar wave functions independently, and afterwards show how the vector 
wave functions hitherto treated are connected to them. 

We have seen that a neutral particle (neutral both as regards charge and 
spin) has an algebraic strain vector ( 6-6); in this case the strain vector S 
becomes a strain scalar. Following the general procedure in wave mechanics, 
we represent it as the product of two scalar wave functions 0^. We may 
expect that scalar wave functions will suffice to represent any macroscopic 
distribution which is neutral as regards charge and spin in fact, such 
distributions as are fully specified by an energy tensor T^ v . If T^ is inade- 
quate and the distribution is of a type represented by a general Riemann- 
Christoffel matrix (11-4), vector wave functions will be required. The 
general theory of the representation of T^ v by scalar wave functions is given 
in 13-7. However, it is not of primary importance to ascertain the precise 
limits of the application of scalar wave functions; the main consideration is 
that they occur in certain problems of great importance in connecting 
relativity theory with quantum theory. 

In macroscopic theory, energy-density, momentum-density and pressure 
are components of an energy tensor T^p . It is part of the definition of these 
quantities that they satisfy the law of conservation (T*P)p = 0; and by this 
property T a p is identified with a geometrical tensor 

-87r/c2 T a/ , = C? a]5 ~j9f aj8 ((?-2A), (13-11) 

which satisfies the law of conservation identically. We insert the constant K 
(constant of gravitation) in order that we may be free to use the unit of 
mass, defined by jp ft = id/dx^, which simplifies the formulae of wave 
mechanics. 

Here A may be any constant, a change of A being a change of the zero from 
which energy and pressure are reckoned. But the zero condition, 2^ = 0, 
should not be taken to correspond to entire absence of matter and radiation 
(or in wave mechanics to zero probability of the presence of particles and 
photons). For if we adopt this as the zero condition everywhere, we are 
comparing the actual universe with a universe completely devoid of matter 



230 Physical Applications [13-1 

and radiation and therefore without observable properties. We cannot adopt 
as a reference standard for physical measurements a system in which all 
such measurements become indeterminate. We shall later consider the 
convention adopted for fixing A in current practice. 

To make a transition to wave mechanics, T^ must be interpreted as the 
expectation value of an operator T a . The only appropriate operation is 
covariant differentiation, and to satisfy tensor conditions we adopt 



where 8 stands for covariant differentiation, and m is an invariant. We shall 
be concerned only with scalar wave functions and natural coordinates; so 
that in our applications the covariant derivatives can be replaced by ordinary 
derivatives. 

No hypothesis is implied in (13-12). It defines the wave functions which 
will be used for the description of macroscopic systems, viz. they are to be 
such as to give at each point an expectation value of T a ^ which agrees with 
the tensor defined in (13-11). The information contained in T^p is thereby 
transferred to a pair of wave functions and made available for treatment by 
the methods of wave mechanics.f 

Equation (13-12) is consistent with the usual definitions of momentum 

p a =-i8/S# a , pv^rndx^ds. (13-131) 

For, substituting in (13-12), we have 

T a)3 = p j 8P a /m, (13-132) 

so that, when the momenta reduce to eigenvalues, 



*. (13-133) 

m da ds v ' 

which is a well-known form of the energy tensor of a system of particles 
when the right-hand side is summed in an appropriate invariant way.J 
Here, however, we regard (13-12) as the fundamental formula, not implying 
analysis into particles with eigenmomenta, but directly translating the 
curvature invariants of macroscopic theory into the operational forms of 
wave mechanics. 

The coefficient m should not be prematurely identified with the proper 
mass of the system represented by the wave functions. As it is of the dimen- 
sions of mass, we may call it the mass-constant of the system. In most 
applications the units are adjusted so that, for an elementary wave system, 
m is the proper mass of the corresponding particle. For a composite wave 

f In 13-7 we shall follow the inverse procedure. The wave functions are defined directly, 
and the operational form T a(3 required to represent T a(3 is deduced. 
J Mathematical Theory of Relativity, equation (53-1). 



13-1] Standing Waves 231 

system m can be interpreted as the proper mass of a single equivalent 
particle which is by no means the same thing as the sum of the masses of 
the particles. For the purposes of general theory the division by m is rather 
inappropriate, and it would be better to write (13-12) as 

wT aj5 = -S*/S&0&c a , (13-125) 

defining a pressure invariant operator mT^p (cf. 12-6). 
The expectation value is formed in the usual way, namely, 

Z!0 = fr^. (13-141) 

Thus tensor conditions are satisfied if <f> and \f> are invariants. For a small 
three-dimensional volume dV 

T^pdV^^T^dV. (13-142) 

If a component T u p reduces to an eigenvalue t^p, we have 

T^pdV^t^p.^dV. (13-143) 

This is conveniently regarded as expressing that the oc/J-component of the 
energy in dV is due to a probability </>^dV that an entity with energy com- 
ponent ^p is within the volume. We call this entity a scalar particle. It will 
be found later that an elementary scalar particle is equivalent to four 
elementary charged particles. 

If the wave functions of the distribution are analysed into a set of elemen- 
tary orthogonal wave functions, each of which is a discrete eigenfunction 
of certain components of T a p (in practice, the diagonal components) 
sufficient in number for the eigenfunction to be defined uniquely by their 
eigenvalues, we can introduce an exclusion principle. The exclusion prin- 
ciple states that in the whole domain of V there is not more than one scalar 
particle for each eigenfunction; so that the whole probability attached to 
any set of eigenvalues is not greater than 1. Using normalised eigen- 

functions $ ny (f> n which satisfy 

* 

(13-144) 

$ n and <f> n will occur in the general wave functions Sa n n , Sa n '^ n with 
coefficients not exceeding 1. Or, if w ' = a w n , ^ w ' = tt n '^ w > 

(13-145) 

The foundation of this exclusion principle will be investigated in Chapter 
xvi. Meanwhile we accept the general idea of such a principle from current 
quantum theory. 

The important point now arises that, since ifi n , <f> n are invariants, the con- 
dition (13-145) is not invariant for Lorentz transformations or other 
relativity transformations which change dV. Thus, if it applies at all, it 



232 Physical Applications [13-1 

applies to a special time axis. The particle interpretation is subject to the 
drawback, that by changing the time axis, the probability of the particle's 
existence in the domain V may become greater than unity ! This has been 
urged as an objection to the use of scalar wave functions; and Schrodinger's 
development of them has been condemned as non-relativistic. Briefly the 
criticism is that, in order that the probability <f>^dV of the particle being in a 
specified three-dimensional volume may be invariant for transformations 
of the coordinates, </>i[i must be a vector in the direction normal to dV. 
Therefore </> and ifi cannot be scalar quantities, but must be wave vectors 
whose product yields a space vector. 

Much of Schrodinger's theory is non-relativistic; but in this instance 
the condemnation is not justified. The criticism overlooks that, whereas 
(13*142) is a tensor equation, invariant for all relativity transformations, 
(13-143) is definitely not invariant for any continuous transformation, since 
it is restricted by the non-invariant condition that $ corresponds to the 
eigenvalue t^. By hypothesis the eigenvalues defining $ are discrete; 
otherwise no question of an exclusion principle arises. If we make an in- 
finitesimal transformation of the coordinate system, the adjacent values of 
t a p are impossible as eigenvalues, and $ does not exist. It is a consequence 
of the discreteness of the eigenvalues that (13-143) can apply only to special 
orientations of the axes. 

The physical reason is evident. Discrete eigenvalues arise from boundary 
conditions; and the components of the energy tensor which are quantised 
are those which have a special relation to the characteristics of the boundary. 
In particular, if the boundary is of constant form, it determines a space-time 
frame with respect to which it is at rest. The time component of the energy 
tensor thus becomes separated from the other components. 

We have said that (for the present) we base the general idea of an exclusion 
principle on current quantum theory. As formulated for atomic systems it is 
amply confirmed by experiment. Undoubtedly an exclusion principle of 
some kind is obeyed by macroscopic distributions, and it is important to 
discover its precise formulation. There has not been the same opportunity 
for experimental test,f and formulae are current which have not been 
quantitatively verified. It is therefore necessary to examine the principle 
critically. 

It is agreed that the unit wave functions (commonly regarded as repre- 
senting individual particles) are eigenfunctions of a certain number of in- 
dependent operators U^, and that the function is determined uniquely 
when the eigenvalues u^ are specified. Also the eigenvalues are discrete, 

f I think that the calculation, which we shall make later, of the constant of gravitation 
and the cosmical constant provides the first quantitative observational test. A qualitative 
test has been provided by the phenomenon of white dwarf stars. 



13-1] Standing Waves 233 

although in some applications they may be very numerous and fall close 
together. It is therefore impossible to vary an eigenstate continuously; 
and if the distribution is given a small velocity, as in a Lorentz transforma- 
tion, it must cease to be an eigenstate. It is therefore a necessary condition 
for an exclusion principle that it shall not satisfy the Lorentz transformation. 
Effectively this means that it applies to the relative or internal coordinates 
of a system as is illustrated in the application to the hydrogen atom. For 
a neutral macroscopic distribution the choice of operators U^ is very limited, 
since the only recognised characteristic of the distribution is T^ . In familiar 
applications <f> and are taken to be eigensymbols of the four diagonal 
components of T^\ so far as 1 can see, the only other possibility would be to 
employ derivatives (more especially curls) of T a p . 

For the most part, what we have said about Lorentz transformations 
applies also to ordinary rotations. But when, owing to the symmetry of a 
system, it is impossible to distinguish its orientation with respect to its 
surroundings a degeneracy occurs; and the discrete set of eigenf unctions 
<f>, ^ is defined by operators U^ which include angular momenta. 

There is no ground for supposing that an exclusion principle applies when 
the distribution is not in a steady state; and it seems evident that it does not 
apply. The energy concentrated in one elementary state of an organ pipe 
may (temporarily) be very much greater than hv ( 13-5). Consequently the 
principle cannot be extended from discrete to continuous wave functions, 
since the latter postulate an unbounded region in which equilibrium cannot 
be realised. When the eigenstates are very close together we may, for mathe- 
matical convenience, treat them as continuous, e.g. the states of a hydrogen 
atom just below ionisation; but this statistical continuity must be dis- 
tinguished from the genuine continuity of the eigenstates which begins just 
above the ionisation level. If the ionised hydrogen atom is in a finite en- 
closure, the free electron can attain a steady probability distribution; but 
the eigenstates have again become discontinuous owing to the boundary 
conditions imposed by the enclosure. 

The exclusion principle is often stated in the form that there is not more 
thanoneelectronperunitcell(A 3 ) of phase space. This is the most convenient 
form for statistical purposes, feut since it no longer explicitly requires the 
wave functions to be discontinuous, many physicists have assumed that it 
applies to systems represented by progressive waves, which are not in 
statistical equilibrium. Any proposal to extend formulae relating to statis- 
tical equilibrium to non-equilibrium conditions requires justification; but 
no such justification has been attempted, and there is neither theoretical 
nor experimental ground for the assumption. 

We add a comment on the association of particles with the unit wave 
functions. We are not attempting to analyse the macroscopic distribution 



234 Physical Applications [13-1 

into the elementary particles, or even the neutral particles, treated in micro- 
scopic theory. Any particles which we now introduce are average particles, 
having composite individuality; and as such they have different properties 
from the elementary particles. It is permissible to regard an electron of 
composite individuality as an electron, in the same sense in which it is per- 
missible to regard the Prime Minister as a human being although he is 
subject to changes of size and appearance impossible in a human being as 
biologically defined, and might undergo displacement from Lossiemouth to 
Bewdley without traversing any intermediate region. An average particle 
is the particle which is fulfilling a particular role; and the conception arises 
when there is in the assemblage always one particle, or a probability of one 
particle, playing that role; but the identity of the particle is continually 
(and, for statistical purposes, continuously) changing. We shall find later 
that it is impossible even in microscopic theory to assign identity to particles 
at different times in ah absolute way. But it is here unimportant whether 
the continuous identity which we assign to a microscopic particle is absolute 
or relative; adopting the microscopic particle as standard, the identity of 
the average particle associated with a macroscopic eigenfunction is com- 
posite and continuously changing. Naturally therefore it does not obey the 
same laws as a microscopic particle, in so far as the laws involve d/dt. Since 
d/dt gives the mass and energy, the mass and hamiltonian are different. 
There are, however, well-known theorems concerning the density of distri- 
bution of eigensolutions, which normally secure that the number of particles 
found in different modes of dissection is invariant. We are therefore able to 
state the number of elementary particles corresponding to a given number 
of average particles. 

13-2. The Box Problem. 

Consider a rectangular block of matter (e.g. in the interior of a star) of 
dimensions Z x x Z 2 x Z 3 , and containing al^LJi^ electrons. We wish to deter- 
mine the minimum electron pressure P as a function of the electron density a . 
We take axes such that the block as a whole is at rest, and we analyse the 
internal state of the block into standing waves in the relative coordinates 
i & fa- This is done by resolving the scalar wave function (within the 
block) into Fourier components, so that ^ ^^n v n v n^ where 



(13-21) 



"2 



n n -n cos -^ COS 

n v n v -n 8 ^ j 



etc., and n l9 n 29 n$ are integers (positive in (13*21)). The factors can be cosines 
or sines, and (as a convention) we distinguish sine factors by negative values 



13-2] Standing Waves 235 

of 7i. The condition governing this analysis is that the wave functions 
defining the elementary states must form a complete set of orthogonal 
functions capable of representing any distribution of $ within the block. 

Let s* a =27ra a /J a , w * = w ^ + w f + tu 3 2 . (13-22) 

Taking w l9 w%, w 3 as rectangular coordinates, each elementary wave is 
represented by a point in ro-space. The points form a rectangular lattice 
with intervals 27r/Z a ; and their density in to-space is / 1 Z 2 / 3 /(27r) 8 . By the 
exclusion principle there are not more than two electrons (and two protons) 
to an elementary scalar wave.f Hence we shall require 0^/2^3 waves, 
which will occupy a minimum volume i (27r) 3 a in ro-space. If this volume 
forms a sphere of radius r, 4^3 = | (27r )3 a . (1 3 . 2 3) 

The pressure is given by the operator 

P = J (T u + T 22 + T 33 ) = - V 2 /3w (13-24) 

by (13-12). For an elementary wave *l>n v n v n^ ^^is reduces to an eigenvalue 



by (13-21) and (13-22). Hence 

Py 2 Z 3 = S w*/3m = oi^Zg .o*/3m, (13-25) 

where ra 2 is the mean value of to 2 . We have therefore to select the waves 
which give the minimum value of w*. This is obtained by packing them in 
the sphere of radius r above mentioned. Then 



(13-26) 

by (13-23). Hence, by (13-25), 

D _l/3\t(fcr)i 



In C.G.S. units the factor (27r) 2 in the numerator is replaced by A 2 , our 
present units being such that A/2?r is unity. (The coefficient of the momen- 
tum operator is A/2?r, not A/TT, since $ is an internal wave function.) The 
constant m is identified with m e . 

This is called the ordinary degeneracy formula. It was first applied by 
R. H. FowlerJ in his investigation of the state of matter in white dwarf stars 
in 1926. But for many years it has been discarded by astronomers in favour 
of a supposed relativistic degeneracy formula. The " relativistic " degeneracy 
formula appears to be without foundation. The difference is important in 
the theory of evolution of white dwarf stars; and it was the paradoxical 
results of the "relativistic" formula, disclosed in an investigation by 
S. Chandrasekhar, which led me to examine its validity. || 

t We take this result from current theory. 

i Monthly Notices, R.A.S. 87, 114. 

Ibid. 95, 207. 

|| Ibid. 95, 194; 96, 20. See also 13-8. 



236 Physical Applications [13-2 

From a theoretical standpoint it is of greater interest to take the material 
to be ionised hydrogen, and calculate the whole pressure including proton 
pressure. The separation of the electron pressure from the proton pressure 
is somewhat artificial, since the electrons could not exist alone. If a is the 
number of scalar particles per unit volume, and ra the mass-constant of a 
scalar particle, i.e. the constant ra in (13-12), the minimum pressure is found 
by the foregoing method to be 



Suppose that I == ^ ==2 ( + ) (13-291) 

^ ra ra \m p mj 

by (12-63). Then (13-28) becomes 



Since the electron density is now 2<j, this agrees with (13-27) supplemented 
by a corresponding proton term. 

We therefore conclude that the coefficient ra for a scalar wave function 
is as given in (13-291). This is in the system of units with A = 2rr. We recall 
that when the relativistic double valued wave vectors are employed, the 
units are such that h = TT ( 9-6). In that case 

ra = 2ra /136. (13-293) 

The factor 136 is a transformation factor due to our normal reckoning of 
mass being determined by double vector wave functions with a phase space 
of 136 dimensions, whereas the double scalar wave function has a phase 
space of one dimension. We could eliminate it by assigning appropriate 
indices to scalar and vector wave functions (8-65). The factor 2 occurs 
because here the internal coordinates f ^ are relative to the centre of mass, 
whereas in the simple transformation (10-911) ^ is the coordinate of one 
particle relative to another. 

13-3. The Energy of Standing Waves. 

The domain of the internal coordinates ^ is three-dimensional, there being 
no relative time coordinate. Consequently T^ for standing waves consists 
of spatial components only. This is not the whole energy tensor; the re- 
mainder is provided by the external wave function which represents the 
motion of the block as a whole. The rest-mass is contained in the external 
wave function. In the present coordinate system (chosen so that the block 
as a whole is at rest) the external energy tensor consists of a single com- 
ponent 2V Since f^T^^T, we have in our coordinate system (with real 

time) Tu-T + Tu + TB + Ta, ( 13 ' 31 ) 

which is also written as / >=p + 3P. (13-32) 



13-3] Standing Waves 237 

Consider now a different set of standing waves in the material of the 
block, so that the pressure P is changed. One or both of the quantities />, p 

must be changed. Let 
8 



(13-33) 

where /> ' is constant, and /? may be zero or constant or a function of P. The 
usual point of view is that it is more appropriate to divide the whole energy 
tensor into a constant external part T^p^, and group together the part 
(P, P, P, (3 + j3) P) which comprises the whole effect of the standing waves. 
To provide for the energy (3 + /?) P we must insert a time factor e iki in the 
functions *ff n t n % n , k being dependent on (n l9 n 2 , n 3 ) or equivalently on 
(w l9 m 2 , iu 3 ). Owing to our special choice of axes t is identical with the 
dynamical coordinate s of the internal state; so it is not so incongruous as 
it might seem, to mix it with the internal coordinates f^. There are no 
boundary conditions for determining k, and it must be determined from the 
energy density (3 + j8) P; but this requires a knowledge of /?. 

The problem of finding the hamiltonian of the standing waves, i.e. the 
expression for the energy operator id fit in terms of the momentum 
operators id/dg^, thus resolves itself into determining the constant or 
function j8. We wish to find the change of energy density or mass density 
85P 44 , when energy is added to the system in the form of standing waves 
producing a pressure ST U . 

It is commonly taken for granted that the answer to this problem can be 
checked by observation. No such test is possible; and any answer we may 
adopt must rest on convention, not observation. Consider a vessel con- 
taining gas. I do not doubt that when the pressure is increased by raising 
the temperature of the gas, the mass (measured in the ordinary way by the 
acceleration of the vessel under a given applied force) is increased by the 
amount of the heat energy that is added. Further, if the gas is monatomic, 
its heat energy is fully represented by standing waves. But for our purpose 
the experiment is illusory. The integrated pressure of the gas is precisely 
balanced by the integrated tension of the walls of the vessel. The observed 
change of mass (integral of 8jP 44 ) is therefore not associated with any net 
change of T U9 but with a differential effect depending on whether T n is in 
a gas or a cohesive solid. The complication in the solid is that there are inter- 
atomic forces of cohesion, of a type which can only be represented by the 
use of vector wave functions. We can make an arbitrary addition ST^ = o&T n 
(a = const.) to the density at each point in the gas and vessel, since it is 
only possible to test observationally systems in which pressures and tensions 
balance. Accordingly, ^8T n = 0; and the addition cancels out on integration 
over the system. 

It may seem more hopeful to examine a steady system without a con- 
straining boundary, e.g. a star cluster, or a star composed of monatomic gas. 



238 Physical Applications [13-3 

But has a star greater or less mass than the sum of the rest masses of its 
particles? It has more kinetic energy but less total energy. Which of these 
corresponds to the mass? Or, more precisely, which is represented by the 
integrated value of 5P 44 ? The answer to the latter question is neither ; for 
jP 44 cannot be integrated in an absolute way in a curved space. In trying 
to avoid a constraining boundary we have used curvature of space-time 
(gravitational force and potential energy) to keep the system steady; and 
the same indeterminacy now appears in the form of non-integrability . 

This observational indeterminacy is provided for in the initial formula 
(13-11). Tp V is indeterminate to the extent of an additive constant Agr^/, and 
the relation of S!F 44 to 8T n will depend on the arbitrary choice of a corre- 
sponding 8A. 

We have therefore to fix the relation between 8jP 44 and 8P by a convention. 
For reasons, which we shall presently explain, the convention is taken to be 

ST = 0, (13-34) 

so that j8= 0. This will fix the change (if any) of the gauge constant A, and 
it can therefore be regarded as a gauging equation. It is here asserted only 
for changes of standing waves; in particular it does not hold for regions in 
which there is unbalanced angular momentum. But it happens that it 
includes Maxwellian electromagnetic fields since these have a proper density 
T which is identically zero. 

The invariant T is the Action of the material system. The action of electro- 
magnetic waves (but not of aperiodic electromagnetic fields) is zero. Hence 
(13-34) secures that there is no change of action when radiation is absorbed 
and converted into standing waves in a material system. It may therefore 
be interpreted as a Principle of Stationary Action for variations of the equi- 
librium state of matter and radiation. It is in the form of an action principle 
that the convention (13-34) has become incorporated fundamentally in the 
current scheme of physics. 

We must therefore accept (13-34) as the current convention. Then j8=0; 
and, by (13-32) and (13-33), the energy density T 44 of matter whose internal 
state is represented by standing waves is 



(13-35) 
where p is constant. 

In the classical theory of gases, and in elementary quantum theory, 
only the kinetic energy, which is approximately fP, is considered; and 
the energy density is taken to be p=/> +fP. But changes of pressure 
cannot be produced without changes of the gravitational field ; and it is 
disastrous to introduce relativistic refinements without taking into account 
the changes of potential energy. 



13-4] Standing Waves 239 

13-4. The Use of an Action Principle. 

Wave mechanics is a statistical theory and its results refer primarily to 
systems in statistical equilibrium. Its procedure is to investigate the pos- 
sible steady distributions of probability, postulating an ideal environment 
of the system considered. Its dynamical equations are derived from the 
condition that a recognisable characteristic of the system remains steady 
this being taken as the criterion of a steady state. Herein lies the essence of 
the statistical method. For the complexions of a system originally regarded as 
distinct are regrouped according to the values of the selected characteristic; 
and the probability distribution of its components takes the place of the pro- 
bability distribution of the original classification. In particular, it is possible 
to find attributes for which, on the original basis of statistics, some values 
are infinitely more probable than others corresponding to singularities 
in the transformation from the old to the new classification of complexions. 

But how can a theory of steady conditions provide anything for obser- 
vation to get a grip on? Those influences from the external world which 
reach our senses are due to change and transition. 

We have made provision for perturbation of and by these steady systems. 
The first step is to introduce steady perturbations. These are found by treating 
the perturbed and perturbing system as a combined system in statistical 
equilibrium, and therefore falling within the scope of the statistical theory. 
We then derive an equivalent representation as two separate systems each 
of which is uniformly perturbing the other. The perturbations are expressed 
as changes of the probability factors attached to the steady states of the 
two systems. 

But it would seem that ultimately there must be some limit to the treat- 
ment of phenomena by methods based on the postulate of statistical equi- 
librium. The universe is far from statistical equilibrium; so that sooner or 
later we are bound to overstep the limits of the theory. I am not sure that 
this conclusion is logically sound. According to our usual outlook the 
universe is far from statistical equilibrium; but it may depend on how we 
choose the basis of statistical enumeration. The "recognisable character- 
istic " J of the universe is that it conforms in every detail to our accumulated 
knowledge of what has actually occurred during a period of a few thousand 
years. We can at least say that J is constant for changes of the dynamical 
time coordinate 8 except in so far as subjective influences (discovery that 
certain information is false) may cause " perturbations". But it is perhaps 
unlikely that the number of symbols U l9 U 2 , ... commuting with J is suffi- 
cient to justify an analogy with the theory of 9-1. 

Be that as it may, physics does not attempt to press the equilibrium theory 
to such an extreme, but breaks off in a new direction. We must make clear 
the nature of this fundamental departure. 



240 Physical Applications [13-4 

A simple illustration is afforded by the historical development of the law 
of gravitation. First the steady states of a combined system the sun and 
a planet were discovered, leading to the formulation of Kepler's laws. 
Then an equivalent representation as two simple systems, one perturbing 
the other, was found; and the perturbations were expressed in the form of 
the inverse-square law of gravitation. The next step was to assume that 
the same law of perturbation applies, whether the bodies form a steady system or 
not. The Newtonian and Einsteinian pictures of gravitation are such that it 
seems pedantic to emphasise the arbitrariness of the last step. What possible 
bearing can the steadiness of the system have on the matter? But our point 
is (1) that an analogous step must always be taken in developing the general 
laws of nature from the study of steady states, and (2) it is an exceedingly 
dangerous kind of generalisation to apply to statistical formulae. We know 
well the many fallacies which have arisen from applying to non-equilibrium 
distributions the laws found for statistical equilibrium. 

We have therefore to recognise that, woven into the method of physics, 
and forming an indispensable part of it, there is a hypothesis or an assump- 
tion or a convention (we leave the appropriate term for further consideration) 
that results obtained for systems in statistical equilibrium can, in certain 
circumstances which must be strictly defined, be applied to non-equilibrium 
systems. 

The hypothesis or assumption or convention is the Principle of Stationary 
Action. This asserts that a certain characteristic (action) of the combined 
system in statistical equilibrium remains stationary for small deviations 
from equilibrium. This gives, as it were, a slight play at the joints of our 
systems, by which we can extricate them from the bondage of statistical 
equilibrium. After analysing a distribution in statistical equilibrium into a 
number of separate, but mutually perturbing, systems, we can give those 
systems a freedom which they did not possess as components of an equi- 
librium distribution. 

The introduction of some such principle is not a wholly arbitrary pro- 
cedure. Some such "loosening of the joints" is inseparable from the con- 
ception of the analysis of a whole into its parts.f It is meaningless to write 
a = b + c unless we contemplate the possibility that b may have a significance 
when c is not added to it. As a condition for detaching b from c, we must 
recognise a definite distribution of the characters of a between 6 and c. 
Thus an important aspect of the principle of stationary action is a localisa- 
tion of the characteristics energy, spin, etc. of the combined system. 

Let us return to the problem of determining observationally the change 

t Of. the introduction of an infinitesimal element of relative time dt r in 12-3. It is there 
postulated that the probability distribution of the combined system in phase space is 
stationary for such a variation. 



13-4] Standing Waves 241 

of energy with pressure of a gas. To render the conditions static and thereby 
amenable to treatment by the statistical theory which contains our funda- 
mental definitions, it was necessary to enclose the gas in a vessel. But the 
enclosure introduced compensating tensions which frustrated us. Having 
introduced the envelope, we were unable to detach the gas from its envelope. 
As an alternative we may consider a small volume of gas in the interior of 
a star, and seek to determine by ideal observations how its energy would be 
changed by a change of pressure. The question is absurd; we cannot change 
the pressure at one point in a star (in steady conditions) without altering the 
whole star. Observation will not tell us what part of the whole change of 
energy is located in the particular volume considered. 

To attach an observational meaning to a local association of energy 
density and pressure, we must be able to produce a change whose effects are 
confined to the locality. Such a change is provided by the conversion of 
radiation into molecular motion of matter. According to (13-34), if the 
radiant energy is converted wholly into standing waves, as in an ideal mon- 
atomic gas, there is no change of energy density or pressure;! 8O that the 
effects are confined to the locality. In particular the gravitational field 
emanating from the region is unaltered, so that there is no cause of readjust- 
ment of the matter outside. But there is the preliminary objection that the 
existence of radiation in any other than its equilibrium proportion pre- 
supposes a highly disturbed state of the star. Thus even the conversion of 
radiation into molecular motion is not a strictly local phenomenon. Its 
effects are local, but its causes are not local. It cannot occur independently 
of a general settling down of the star, implying similar conversions in other 
regions. 

It is here that the principle of stationary action, which must now have 
the definite form (13-34), steps in. It asserts that 8 T = applies to the local 
conversion of radiation into material energy, independently of the con- 
versions occurring in other parts of the star. Although the rest of the star 
could not actually be in statistical equilibrium, it may be treated as if it 
were in statistical equilibrium; because the property with which we are 
concerned is stationary for small deviations from statistical equilibrium. 

The mathematical form of the action principle shows quite explicitly that 
it is a means of localising the characteristics of the universe. The quantity 
to be varied is an integrated quantity covering a large volume; the quan- 

t This appears to be inconsistent with the elementary formulae, which predict a change 
of pressure if the volume is unchanged; hut in our problem we have to admit whatever change 
of volume is necessary in order that the effect of conversion may be strictly localised, i.e. 
that there may be no change of the gravitational field outside the region. But a real change 
of volume would displace the surrounding matter outwards or inwards; the change must 
therefore be represented as a change of reckoning of volume, implying a change SA of the 
gauge-constant. 

ETP 16 



242 Physical Applications [13-4 

titles determined or defined by the variation are functions of position so 
that each minute part of the system is described separately. Localisation is 
an artificial conception in an interrelated universe, where the influence of any 
one part extends through the whole. To detach one part from the rest is 
impossible. The action principle reassures us by asserting that (with suitable 
safeguards) a mild impossibility is a permissible idealisation. Its ill effects 
are of the second order of small quantities. 

Like many of the "principles" in science, the genesis of the action prin- 
ciple is that, having realised that we ought not to make a tempting assump- 
tion, we erect a principle to say that we may make it. I shall not attempt to 
defend the principle of stationary action as a physical assumption; there is 
no need to do that. Its defence is that it is the basis of our definitions. A 
principle of localisation must precede the definition of any localised entity, 
e.g. the g^ v or F^ of field theory. By definitions I here mean, not the mathe- 
matical definitions as symbols in a deductive theory, but the definitions by 
which they are recognised and measured in observational science. The action 
principle is not a physical hypothesis; it is a means of defining localised 
quantities. Thus in the application in 13*3 we had an undefined disposible 
constant A. By adopting the action principle ( 1 3- 34) we remove the arbitrari- 
ness of 8A, and changes of energy and pressure are defined precisely. 

In considering the action principle as a vehicle of definition, it is desirable 
to distinguish between : 

(a) Weak action principles. Referring to conditions in empty space, with 
or without weak electromagnetic fields. 

(6) Strong action principles. Referring to conditions in continuous matter, 
intense electromagnetic fields, the interior of a nucleus or electron, etc. 

The difference arises because, although an investigator has a certain 
amount of freedom in adopting definitions of the terms which he employs, 
he must have regard to established usage. In regard to (a) established usage 
dictates the form of the action principle. There is no difference of opinion 
as to how the kernel of the action invariant is constituted, to the order of 
accuracy required in weak conditions; for it has to lead to definitions already 
recognised. In regard to (6) an investigator has entire freedom in his choice 
of action invariant and consequent definitions of g^ v and F^, provided only 
that it converges to the weak action invariant as weak conditions are 
approached. Current literature shows that full advantage is taken of this 
freedom ! 

Observational measurements are only made under weak conditions. The 
"distance between two points" in field-free conditions is what it is deter- 
mined to be by measurement; the distance between two points in an intense 
magnetic field is whatever (within reason) a theorist chooses to call it. No 



13-4] Standing Waves 243 

experimental physicist would attempt accurate measurement of distance 
in an intense magnetic field; he would mistrust his scales or other apparatus. 
In practice he does not measure the curvature of the track of an electron 
in an intense magnetic field; he measures the curvature of a track on a 
photographic plate placed outside the field. Intervals, and the corresponding 
g^, are only defined observationally in field-free conditions, or when theory 
(that is to say, the accepted action invariant) assures us that the electro- 
magnetic field is not strong enough to introduce sensible error. When once 
we begin to suspect that our measured distances need correction, we turn 
to theory to tell us what the measures ought to have been. Naturally the 
corrected measures will confirm the theory used for determining the cor- 
rections, whatever form that theory may take. 

The literature of mathematical physics abounds in proposals of universal 
action principles covering strong as well as weak conditions. I think it is not 
unfair to summarise the majority of such proposals as follows: 

(1) A geometry based on a new set of axioms is outlined. 

(2) A fundamental invariant of the new geometry is selected as action 
invariant. 

(3) It is shown that the action principle leads to the usual equations for 
weak fields. 

(4) Second order terms, sensible in strong electromagnetic fields, are 
found, which it is suggested may provide an observational test of the theory. 

(5) A hope that something further may come of it. 

Our comment is: (1) We may choose any kind of space we please for the 
purpose of graphical representation of physical quantities without com- 
mitting ourselves to any thing, f (2) The quantities g^, Jf^ are to be used in 
strong conditions in which there is no agreed definition as to how they are 
to be measured. The action invariant embodies the definition selected in the 
new theory. (3) This shows that the definition is not in conflict with any 
established usage. (4) The result of the test will inevitably be a triumphant 
verification of the new theory, provided that the observations are correctly 
reduced. By correctly we mean that, for measurements with apparatus 
actually located in the intense fields, the readings of the apparatus (which 
are, of course, affected by the field) are corrected to accord with the readings 
of ideal apparatus which conforms to the equations of the new theory; and 
for measurements made outside the fields, the inference as to what is 
happening within the field is calculated according to the equations of the 
new theory. 

The trouble about unified field theories is that there are so many of them, 
and all of them are right. The various action principles are various plans of 
t Mathematical Theory of Relativity, 83. 

lC-2 



244 Physical Applications [13-4 

localising the characteristics of the interrelated conditions which the 
universe presents to us. The procedure of localising an entity is inseparable 
from the procedure of defining a local entity. It all reduces to a question of 
definitions. The difference between the rival theories really rests on (5) 
as to which opinions will naturally differ. 

13-5. Potential Energy. 

Let us return to the problem of the rectangular block. In order to avoid 
inessential complications, we shall at first suppose the density to be very 
low, so that collisions of the particles can be neglected. From one aspect the 
block is occupied by the set of standing waves introduced in 13-2. From 
another aspect it is occupied by n elementary particles moving in diverse 
directions; these are represented by progressive waves which combine into 
an w-tuple wave function. I think it will save much confusion of thought if 
we notice that the standing wave function represents a system in equi- 
librium, and the n-tuple wave function represents a rapidly dispersing 
system. If we insert in the n-tuple wave function a later value of the time, 
it gives the distribution which would be reached if each particle continued 
to move with its present momentum. 

Let us then take a somewhat modified ra-tuple wave function Y, which at 
every time t represents the n particles occupying the block at that time. 
This will represent a statistically steady system, just as the standing wave 
function does. Every time a particle crosses the boundary T changes dis- 
continuously.t We must replace the discontinuities by continuous change; 
otherwise the function is not differentiate and is useless for computing 
energy. But the energy id/dt of this smoothed wave function will be 
altogether different from that of the instantaneous n- tuple wave function. 

The general effect of smoothing out the discontinuities can be seen by 
considering the combined wave function of the rectangular volume A and 
the surrounding matter B which holds it in equilibrium. The steady state 
(represented by standing waves) results from the fact that on the average 
for every particle passing from A to B 9 a similar particle passes from B to A . 
In the combined wave function of A and B this is merely a nominal inter- 
change. The coordinates x p of the pth particle of system A have become 
inappropriate to system A, and the coordinates x q r of the qth particle of 
system B have become inappropriate to system B\ we therefore re-label the 
particle at x p as the qth particle of system B, and the particle at x q ' as the 
pth particle of system A. By the well-known Fermi-Dirac rule this inter- 
change of labels reverses the sign of the wave function in this case the 

t Since we are supposed to have approximate knowledge of the momenta of the particles, 
we cannot know their positions exactly. Thus the time of crossing the boundary is indefinite 
to a slight extent; and the change of T though erratic is not strictly discontinuous. 



13-5] Standing Waves 245 

combined wave function of A and B. In time-averaging the succession of 
reversals of sign is replaced by a continuous factor e'^', which reverses 
the sign at regular intervals irjp v . When we form the energy -i9/9f, 
the factor e ip ** gives an additional energy p v . This interchange energy 
does not appear in the instantaneous n-tuple wave function which repre- 
sents a dispersing system; but it belongs inalienably to the steady system 
represented by the modified function Y or alternatively by standing wave 
functions. 

Interchange energy is evidently potential energy. (I may add that so far 
as we know all potential energy is interchange energy.) The system occupy- 
ing the rectangular block may be kept in equilibrium, either by interchange 
of particles at the boundary, as described above, or by a force exerted by the 
matter beyond the boundary turning back the particles as they reach it. 
But the two processes are the same. For, since the particles are indistin- 
guishable individually, it is meaningless to discriminate between an old 
particle being turned back and a new particle coming in. Thus the inter- 
change effect and the force are identical. If we happen to take one view we 
call the energy interchange energy; if we take the other view we call it 
potential energy in the field of force. 

The rectangular block, which we have been contemplating, must not be 
too small to be treated macroscopically. In many applications the free path 
of the particles will be small compared with 1 19 1 2 , 1 B . The collisions are then 
the main instrument in preventing the particles from straying outside the 
block; the reduced number which hover about the boundary are deemed to 
be turned back by interchange as before. We shall find in Chapter xv that 
the electrical forces controlling the encounters of protons or electrons are 
attributable to interchange. For our purposes it is unimportant whether 
the potential energy included in !T 44 is due to collision interchange or to 
boundary interchange. The formal difference is that collision interchange is 
provided for in the equations and definitions of electromagnetic theory, so 
that it is an intrinsic property of electrons and protons as currently defined; 
boundary interchange is an averaging adjustment which enables us to treat 
a dispersing system as a static system. We may say that when the free path 
is small compared with a macroscopic volume element, the volume element 
is genuinely in equilibrium; when the free path is long, it can be treated as in 
equilibrium. 

When the velocities of the particles are not too great, and at the same time 
the density is too low to give rise to degeneracy, the classical theory applies 
and the density of the kinetic energy is f P. To make up the total p + 3P, the 
potential energy must be f P. Thus the kinetic and potential energies are 
equal. The fact is that the waves represented by scalar wave functions are 
the ordinary elastic vibrations or sound waves of the material. The only 



246 Physical Applications [13-5 

point at which we go beyond classical dynamics is when we apply the exclu- 
sion principle. 

The exclusion principle sets an upper limit to the amplitude of any 
particular wave, such that its energy cannot exceed one quantum. In non- 
equilibrium conditions the amplitude of a particular wave can greatly exceed 
this limit, e.g. when an organ pipe is sounding its fundamental note. This 
emphasises the fact that there is no justification for applying the exclusion 
principle in non-equilibrium conditions. It is intimately connected with the 
interchange forces which maintain equilibrium, as will be seen in Chapter xv. 

In extreme conditions the kinetic and potential energies are no longer 
equal. For constant density, it appears that the energy 3P is wholly poten- 
tial at absolute zero (complete degeneracy) and tends to become wholly 
kinetic at very high temperatures. We should first make sure that this 
separation into kinetic and potential energy has an observational meaning. 
Certain phenomena, e.g. the rate of disruption of nuclei by protons, evi- 
dently depend on the kinetic energy of the elementary particles (not the 
composite average particles); and since the rate of disruption increases 
rapidly with the speed of the protons, the distribution law of kinetic energies 
of the individual protons is involved, and doubtless will in due time be 
ascertained from observations of this kind. The problem of finding this 
distribution law theoretically should be soluble; but apparently it has not 
yet been solved. The individual protons and electrons correspond to pro- 
gressive waves; the present investigation treats only the composite particles 
represented by standing waves, and is not relevant. 

Reference must be made to the current formulae (associated with the 
"relativistic " degeneracy theory) which profess to give the distribution law 
of the kinetic energies and momenta of the individual protons and electrons 
at all temperatures. So far as I know, they may be right at high tem- 
peratures; but it is impossible to trust them, not only because they rest on 
a false conception of relativity, but because they are clearly wrong at low 
temperatures. In these investigations the minimum pressure P which 
corresponds to absolute zero is attributed to kinetic energy of the elementary 
particles. If that were so the protons would, if the density were great enough, 
still have large velocities at absolute zero, and continue to disrupt the 
atomic nuclei. This surely is a contradiction of thermodynamical principles. 
Our own investigation does not discriminate between kinetic and potential 
energies; but coupled with the general physical principle that processes of 
the nature of ionisation or transmutation must cease at absolute zero, we 
infer that the energy 3P must be wholly potential. 

It may be urged in defence of the current formulae that the energies and 
momenta to which they refer are not to be taken as the energies and 
momenta for the purpose of calculating collision effects that they are not 



13-6] Standing Waves 247 

to be interpreted according to the classical conception of motion. To this 
we reply: (1) Then the distinction between kinetic and potential energy 
loses all observational meaning; and we lose nothing by treating them 
together in our own theory. But the question still remains unanswered, 
What is the distribution law of the energies and momenta concerned in 
collision effects? (2) Our criticism of the current theory is that it does 
interpret the energy |P according to the classical conception of motion; it 
treats the corresponding velocity or momentum as a vector to which Lorentz 
transformations may be applied, and thereby introduces change of mass with 
velocity. If the particle has one velocity for the purpose of Lorentz trans- 
formations and another velocity for the purpose of collision phenomena, 
what becomes of the principle of relativity? 

13-6. Transition to Vector Wave Functions. 

b 

For the problem of the rectangular block we have used a form of wave 
mechanics much simpler than that which has been developed in previous 
chapters for the treatment of microscopic problems. If it is asked why the 
simple form does not apply to an atom, we answer that it does not even 
apply to a macroscopic distribution of matter in a smooth spherical vessel. 
In addition to the modification of the elementary orthogonal functions 
^n lf w 2 ,w 8 to correspond to a spherical boundary, a new feature appears. 
There are steady states in which the distribution has a resultant angular 
momentum. 

With a spherical boundary unidirectional motion is no longer inconsistent 
with a steady state. In rectangular coordinates there must be on the 
average as much motion in one direction as in the other direction of f a , and 
the state of motion is accordingly represented by standing waves; but in 
polar coordinates it is not necessary that there should be as much motion 
in one direction as in the other direction of 0, so that we may have pro- 
gressive waves in an angular coordinate 0. Such waves are not essentially 
different from the progressive waves representing a particle moving freely 
in space, i.e. revolving unidirectionally about the centre of curvature of 
space. 

The advent of progressive waves into a problem depends on the existence 
of relativity transformations of the coordinates, which occasion a degeneracy 
of the ordinary steady state solutions. In the problem of the rectangular 
block there are no relativity transformations of the internal coordinates 
M ; taking account of boundary conditions, no other axes are "equivalent" 
to those which are parallel to the edges of the block. In the spherical problem 
all orientations of the internal rectangular axes are equivalent; so that 
rotations in the three coordinate planes are relativity rotations. The time 
direction must be chosen so that the spherical boundary is at rest; it is 



248 Physical Applications [13-6 

therefore still unique, and there are no Lorentz transformations of the 
internal coordinates. 

The machinery for dealing with relativity transformations has been 
developed in our early chapters; and in order to provide for the relativity 
transformations of the internal coordinates we must introduce it into the 
internal wave functions. But for three-dimensional relativity it is un- 
necessary to resort to Dirac's fourfold wave vectors. We can take $, <f> to be 
wave vectors with two components (2-vectors), and associate the coordinate 
planes with Pauli matrices 1? 2 , 3 . As explained in 3-8, 19 2 , 3 , i form 
a minor complete set, and are a simplified representation of the symbols 
E 12 , E 23 , EM, E 19 when the other symbols are not being used. 

Considering an elementary scalar wave, let a be the particle density, or 
the probability that the particle belonging to the wave is within a unit 
volume at the point considered. If 0, <f> are normalised in accordance with 
the exclusion principle (13-144), <r = 0<. Thus scalar wave functions relate 
to the special case in which the stream strain vector /S = ^&* consists of a 
single algebraic component a. But when $ and are 2-vectors, an algebraic 
quantity a cannot be a simple product ^*; a is the sum of two pure com- 

P nent8 



where L is the Pauli matrix associated with an arbitrary plane. The two 
expressions in (13-61) are idempotent when normalised (so that the halfspur 
= ); hence they can be factorised ( 5-6). 

Accordingly the "particle" corresponding to an elementary scalar wave 
is replaced by two sub-particles with the non-algebraic stream vectors 
(13-61). The components Jo- and ^icr^ correspond to energy and angular 
momentum. Thus the sub-particles have opposite spin in the ^ plane. When 
the boundary conditions are such that the angular momentum in a steady 
state is zero, the oppositely spinning particles are constrained to occur 
together; they can be regarded as together constituting an indivisible unit 
a scalar particle; and we are only concerned with the sum of their strain 
vectors which, being an algebraic quantity, is represented by scalar wave 
functions. But when, as in the spherical problem, integrals of angular 
momentum exist, the oppositely spinning particles have independent 
probability distributions and are represented by separate 2-vector wave 
functions. 

The analysis of a system into elementary states has to be considered in 
conjunction with the perturbations anticipated. The elementary state is a 
unit which requires a separate probability factor in perturbation theory. 
In earlier chapters we have had in mind well-isolated systems, whose reaction 
with other systems is limited to weak or occasional perturbations. But in 
these macroscopic problems the "system" described by the wave functions 



13-6] Standing Waves 249 

is usually a volume more or less arbitrarily carved out from its surroundings, 
and the viscous forcesf at the boundary constitute a strong permanent 
perturbing influence. We have different elementary states of a gas con- 
tained in a spherical vessel according as thesphere is rough or smooth, because 
in the rough sphere the permanent perturbation is such as to prevent any 
persistence of probability in one direction of spin as compared with the 
other; there is no occasion to give them separate probability factors. 

The choice of elementary states depends on the approximate rather than 
on the exact boundary conditions. A vessel cannot be perfectly smooth; and 
if we paid attention to the exact boundary conditions, we should conclude 
(correctly) that the only state of statistical equilibrium was one of zero 
angular momentum. But the gas may take a long while to reach this state; 
and we learn more by utilising the complete series of states in an ideally 
smooth vessel, treating the small friction as a perturbation which slowly 
transfers the excess probability of one direction of spin to the opposite 
direction. 

The problem of a smooth spherical vessel is a half-way stage between the 
rectangular block and a particle moving freely in space-time. For the free 
particle the relativity transformations are extended to include Lorentz 
transformations. This additional degeneracy must be provided for by a 
second set of Pauli matrices 0^, commuting with the ^. The two com- 
ponents of opposite spin ^<r(li^) are in turn analysed into two com- 
ponents of opposite electrical sign 

ior(lig(li^). (13-62) 

The wave vectors are now 4-vectors, and the double set of Pauli matrices 
is more conveniently replaced by a set of Dirac matrices with the same 
commutative relations (3-8). The four components (13-62) then become 
the spectral components given in (6-64). 

It is of interest to consider whether (13-62) has any application to macro- 
scopic systems. We might start with 0^ instead of ^, and divide the scalar 
density a into two components of opposite electrical sign Jo*(l tfljj. The 
scalar wave function postulates that these are constrained to have equal 
probability so that they need not be considered separately. Can the 
boundary conditions be modified so that they have independent probabili- 
ties? I think so. The necessary condition is that the system shall be in- 
sulated. With an insulated boundary, there are steady states having an 
excess of components of one electrical sign; just as with a smooth spherical 
boundary there are steady states having an excess of components with one 
direction of spin. 

t The boundary pressure (interchange force) is allowed for by including potential energy 
in the system, so that we avoid treating it as an extraneous perturbation. 



250 Physical Applications [13-6 

If the substance is permanently magnetisable it is also possible to have 
steady states with a preponderance in one direction of the magnetic stream 
vector. 

These states of electric or magnetic excess may also be induced by 
permanent perturbation from outside, i.e. by steady electric or magnetic 
fields. Thus the general problem of statistical equilibrium of a macroscopic 
system, when the angular momentum is not constrained to be zero and 
electric and magnetic fields are not excluded, will involve Dirac's vector 
wave functions. 

When the spins and charges are not balanced, the state of the material is 
only imperfectly described by the ordinary energy tensor. For a full 
description we require the Riemann-Christoffel matrix ( 11-4). But it should 
be remembered that the electric and magnetic fields which occur in practice 
in macroscopic systems correspond to an entirely trivial excess of com- 
ponents of one sign. So that whilst these phenomena are important on their 
own account, there is very little to represent them in the statistics. It is 
therefore more suitable for ordinary purposes to represent the electro- 
magnetic field separately by potential theory, rather than to merge it in 
the general statistical formulation where it would be almost lost. 

If we apply the energy operator (13- 12) to vector wave functions, the 
covariant differentiation will introduce matrices ( 8-3). Each component of 
the energy tensor therefore becomes a matrix, and we obtain altogether 
256 components as when the Riemann-Christoffel matrix is used. It appears 
therefore that the theory of the energy of distributions which involve vector 
wave functions, treated in Chapter xi, could be reached by starting with the 
energy operator (13-12). 

13-7. Origin of the Energy Operator.f 

In this section we shall obtain the formal connection between the energy 



and the energy operator 

T aj8 

By contracting the energy tensor we have 

T = (1/8 
Using this to eliminate A, we obtain J 



t As this section is occupied with ordinary tensor calculus the summation convention is 
used unrestrictedly. 
J I was led to consider #p J0 a p# through correspondence with Dr J. Ghosh. 



13-7] Standing Waves 251 

Consider the gauge-invariant curvature tensorf 

*#tf = #*/*- 3 (*J0 + (^ 
which is invariant for the transformation 

9*fi=Pffafi> K a ' = * a + a0/a* a , (13-722) 

where j8 is any scalar function of the coordinates, and 

= logj8. (13-723) 

Multiplying (13-721) by gr a #, we obtain 

*G=a-6(K*) e + 6K K e . (13-724) 

Hence ^~^^ = *^-^*^ + 2 K^-^ a j3^^ ) 

+ 2{(Oj-iM JC *)> + * l *' (13-725) 
where ^ aj8 = (/c a )^~(/c ]3 ) a . 

Take * a =0 initially; and let A denote an increment produced by the 
gauge transformation. Then, since *# a fa/ap* & + &&$ is invariant for the 
transformation, (13-725) gives 

Wtf-tartW-WJt-tartW + WJfi-tatfP).}, (13-731) 
where a denotes dO/dx^. 

If 8 denotes covariant differentiation 

l 
p 

so that (O/J-^W&fcfcgjB-fyfl.. (13-732) 

(i3>74) 

so that (0J j8 + d a ^=-47rK.)8- 1 T a/ ,/9 (13-751) 

and, by contraction, (0 e ) e + fl0, = - 4 . jS^T/S. (13-752) 

Then, by (13-71), (13-731), (13-751), (13-752), 

A(? T aj8 -|gr a ^) = ]8- 1 (T^-i^T) j 8. (13-753) 

We now introduce wave functions 

*-jB, X = J8-S ' (13-76) 

representing an addition to, or modification of, the distribution whose 
energy tensor is T^ . Our result shows that instead of expressing the modi- 
fication as a change of curvature of space-time (which would involve altering 
the metrical tensor gr aj8 to jS 2 ^), we can express the change of T^ ^g^T 
as the expectation value of an operator T a ^-|gr a ^T, the operator being 
defined by (13-74). 

This is the most primitive connection between the wave functions and 

t Mathematical Theory of Relativity, equation (87-5); the suffixes after the brackets denote 
covariant differentiation. Although this tensor is generally studied in connection with Weyl's 
geometry, we do not here apply it in that way. We use only the well-known analytical 
property of invariance of the tensor. 




252 Physical Applications [13-7 

operational momenta of wave mechanics and the curvature tensors which 
represent momenta, etc., in general relativity theory. 

Equation (13-753) can be resolved into separate equations: 

Afy= X T^ f Ato*T)-j^. x Tfc (13-77) 

provided that the change AA of the arbitrary constant A is suitably chosen. 
The two equations are consistent since 

A((^T) = A(<7 a ^ 

<7 a 0gr being unaltered by change of gauge. The second equation of (13-77) 
is therefore derivable from the first. 

The wave functions employed in practical applications of wave- 
mechanics are an adaptation of these primitive wave functions to specially 
simple conditions. In particular a modified energy operator is used, defined 
by V 32 

' (13 ' 78) 



whose eigenvalue is the integrated energy over a three-dimensional domain 
of volume V. Instead of ^r, x we take two equal wave functions 

A=H0. 

The product \^<j> bears the same kind of relation to ^^ that a strain vector 
bears to a space vector. Instead of (13-77) we have 



If now <f> and \f> are normalised so that f<f>ifidV= 1, we have 



and V . A T^ is regarded as the energy of a single particle distributed through- 
out V with a probability density <fn/t. 

Further light on the connection between wave functions and gauge 
transformations will be found in 14-1, where a special case is treated. 

Although we do not normally employ Weyl's representation of the 
electromagnetic field in wave mechanics, it is of some interest to extend the 
foregoing investigation to his theory. The difference is that we can no longer 
put * a = initially, since the electromagnetic force JP a ^, which is invariant 
for the gauge transformation, is represented by curl /c a . The term 



in (13-725), being non-linear, will give cross terms 

2( 
or, in operational form, 



), (13-79) 

representing a mutual energy of the waves and the electromagnetic field, 
which is not included in (13-74). 



13-8] Standing Waves 253 

13*8. The Degeneracy Formula. 

Since the formula (13*27) for the minimum pressure differs from the so-called 
relativistic degeneracy formula which has been widely but uncritically 
accepted, we shall here explain the origin of the difference. 

If a particle of proper mass m has a velocity vector dx^/ds, its "integrated 
energy tensor" is defined to be 

Uv>-i&^-. (13-811) 

ds da v ' 

Suppose that there is a discrete set of values of UP V , and that there are n r 
particles with energy Iff in a three-dimensional volume V. A fractional 
value of n r is to be interpreted as a probability. The resultant energy tensor 
(reckoned as usual per unit volume) is 

Tfi v = Z> r n r Us v IV. (13-812) 

Let N r = n r (dtlds) r . (13-82) 

Then N r /V is the proper density of distribution of the particles with suffix r 
as contrasted with the relative density n f /V. The energy tensor can also 



where M 

The exclusion principle provides an upper limit to the number n r or N r . 
The difference between the two theories is 

Present theory n r < 1, j 

Current theory N r ^l.]~ ( ' ' 

It will be seen that, if V = 1, the maximum contribution of any one state to 
TP V is on the present theory m (dx^/ds) (dx v /ds), and on the current theory 
m (dXp/ds) (dx v /dt); or in terms of momenta the contributions are ppp v /m 
and P^P V !PQ . In particular the contributions to the energy are, respectively, 



It is clear that N r has been used in current theory under the impression 
that (13-812) is a tensor equation. If each term on the right-hand side were 
a tensor, n r /V would be invariant for Lorentz transformations, and N r would 
be invariant. But the eigenvalues Uf of a tensor operator do not constitute 
a tensor, and (13-812) is not transformable. 

When once it is realised that the calculation of P is concerned with the 
random internal motions relative to the centre of gravity of the whole 
block, so that the time-axis is prescribed, it is difficult to see any reason at 
all for the introduction of N r . For standing waves the momentum does not 
reduce to an eigenvalue, so that the number N r does not exist. 

Since N r m=n r M, the substitution of N r for n r is from one point of view 
equivalent to introducing change of mass with velocity. The pressure in a 



254 Physical Applications [13-8 

star is of order of magnitude p(f>, where <f> is the gravitation potential measured 
from the boundary. If all the particles are of the same mass, v 2 is of the same 
order as ^; and the change of mass with velocity |mv 2 /c 2 is of the same order 
of magnitude as the gravitational potential energy m^/c 2 , whose effect on 
the mass is neglected. It is, however, assumed in the current theory that, 
although v 2 is of order <f> for protons and ions, it is very much higher for 
electrons, which thereby contribute the bulk of the pressure. But there is 
no reason to suppose that in the degenerate state the electrons have higher 
speeds than the protons. Equipartition of energy refers only to transferable 
energy. 

Investigators seem to have been misled by trusting to the classical picture 
of moving particles. When a particle is forced up into a state of high energy 
by the occupation of the lower states, the current theory pictures the energy 
in the classical way as translation with high velocity. It accepts this so 
literally that it supposes that the particle could be reduced to rest by a 
Lorentz transformation, and thereby calculates its change of mass with 
velocity. But non-transferable energy cannot logically be represented that 
way. We take it that the degeneracy energy is potential energy. There is 
then no correction for change of mass with the (non-existent) velocity; and 
the exclusion principle must be n r < 1, since N r is undefined. 

The exclusion principle was first recognised in the atom where it applies 
to the internal motion of a system in a steady state. In generalising it to 
macroscopic systems it would be unsafe to disregard this restriction. In 
any case the problem to which we here apply it relates to an internal system 
in a steady state. If we are not obsessed with the idea that the formulae must 
somehow be capable of extension to dispersing systems represented by plane 
progressive waves, or that they must be invariant for transformations of 
axes which would change the nature of the problem, the procedure is 
straightforward and unambiguous. The steady state distribution is repre- 
sented by standing waves. The choice between n r and N r is settled by the 
fact that N r is not only irrelevant but non-algebraic. As shown in 13-5 the 
steady condition involves a potential energy due to interchange, and for 
complete degeneracy the energy is wholly potential. Finally, we shall be 
able to test the theory in Chapters xiv and xvi, by using it to calculate the 
constant of gravitation and the cosmical constant. 

Originally my choice of n r was derived from the investigation in 10-9, 
where it was shown that the relative energy of a particle is not p Q but 
a ( jpj 2 +p 2 2 +2> 3 2 )/m. Since the law connecting pressure and density in a gas 
depends only on the internal motions, which must be separated from the 
mass motion of the gas as a whole, I could not feel satisfied with the current 
theory which took the energy contribution due to one occupied state to 



13-8] Standing Waves 255 

The present theory gives P Q oc or* at all densities. The current theory gives 
P oc / (a), where/ (a) changes gradually from or* to <r* as the density increases. 
This is especially important in the theory of white dwarf stars (in which the 
degeneracy pressure becomes large), because it can be shown rigorously 
that the pressure in a star cannot exceed Bp*, where B is a constant depend- 
ing only on the star's mass.f The curve P = Kv* giving a minimum pressure 
must, as a and p increase, eventually cross the curve P=JB/>^ giving a 
maximum pressure. The crossing point is an extreme upper limit to the 
density in a star of given mass; this limit is of order 10 7 gm. cm.- 3 for the 
sun's mass. The current theory on the other hand gives an upper limit to p 
only in the smaller stars; above a certain mass the curves no longer cross, 
and it would seem that as the star's energy supply gives out, it must go on 
contracting to ever higher density until the space becomes so much curved 
that the terms "contraction" and "density" lose all meaning. 

t Monthly Notices, R.A.S. 91, 444 (1931). We may take the density p to be an approxi 
mately known multiple of the electron density a. 



CHAPTER XIV 
THE COSMICAL PROBLEM 

14-1. Waves and Curvature. 

Without appealing to the detailed investigations in Chapters xi and xin, it 
is evident that the curvature of space-time introduced in relativity theory 
and the waves of "0" introduced in wave mechanics are equivalent. Both 
devices are used for the same purpose, to represent the distribution of mass 
and momentum of physical systems. Both are devices] it is not suggested 
that either the curvature or the waves exist in a literal objective sense. The 
method of waves gives the finer analysis, and is essential if we are treating 
systems on the atomic scale; but for the mechanics of ordinary macroscopic 
objects, and for astronomical and cosmical systems, either method is 
available, and either method should give the same result. 

But the result will be expressed in terms of different natural constants. 
When the relativity method is used, the principal constant involved is the 
gravitational constant /c. Wave mechanics does not introduce K, but uses 
Planck's constant h and the masses m p , m e of elementary particles. If we 
can find one problem tractable enough to be solved by both methods, we 
shall, by comparing the two answers, obtain a relation between the natural 
constants. I cannot but think that the realisation that a hitherto un- 
recognised relation exists that there will be at least one redundant con- 
stant when the theories are brought together is scarcely less important 
than the ascertainment of its precise numerical form. 

In this chapter we shall solve one problem by both methods, and so 
ascertain the relation between the natural constants. The system which we 
shall consider is a self-contained static distribution of material particles 
(protons and electrons) without radiation. By "self-contained" we mean 
that it requires nothing external to itself to hold it in equilibrium. In 
relativity theory the only completely static self-contained system is an 
Einstein universe. In quantum theory a system, which is static and radia- 
tionless, must be in its ground state. We shall therefore treat an Einstein 
universe, (a) by the ordinary relativity theory, and (6) by wave mechanics 
applied to a system of particles in the ground state. 

As in the preceding chapter, we represent matter in equilibrium by 
standing waves. But there will now be no artificial boundaries, and the 
waves extend throughout the universe. 

In stereographic coordinates the line element of an Einstein universe isf 
<fo2 = -(i+ r */4R*)-*(dx*+dy* + dz*) + dt* (14-11) 

t Tolman, Relativity, Thermodynamics and Cosmology, equations (138-4), (139-3), (139-4). 



14-1] The. Cosmical Problem 257 

and the pressure and density are found by the usual method to be 



(14-13) 

Instead of treating jR as a radius of curvature, we treat ft as a gauge factor. 
That is to say, we treat the stereographic projection as the true configuration ; 
the curvature is abolished, and #, y, z are rectangular coordinates in a flat 
space. The "true" element of length is then ids = (dx 2 + dy 2 +dz*)l, which 
is j8 times the measured length ids. Hence the true volume is j3 3 times the 
measured volume, and the true particle density a is /?~ 3 times the measured 
particle density a. The measured density of distribution of particles in an 
Einstein universe is uniform; in the new reckoning this uniform density a 
is replaced by GP O = /J~ 3 <7. 

Expressing the particle density cr in the flat space as the product of two 
equal scalar wave functions 0, 0, we have 



Prom this we calculate the eigenvalue of V 2 at the origin 

V 2 =-9/4K 2 (14-141) 

so that, by (13-24), P = %lR*m. (14-142) 

To obtain the expectation value P, we must multiply the eigenvalue (at the 
origin) by the probability that the entity represented by 0, is in unit 
volume at the origin. The whole volume of space in natural measure is 

F=27r 2 J2 3 . (14-143) 

Since the distribution is uniform, the probability associated with any unit 
natural volume is 1/F. At the origin j8= 1, and the two reckonings of volume 
agree. Hence p8/4RmF, P = 9/4# 2 mF, (14-144) 

by (13-35). 

Since we have represented the whole density cr by one pair of wave func- 
tions, we have treated it as the probability distribution of a single scalar 
particle. The formulae may be regarded as referring to an ideal one-particle 
universe; or, if applied to the actual universe, m is the mass-constant of the 
equivalent single particle. The extension to a system consisting of N 
particles which obey the exclusion principle will be treated in 14-2. 

In Chapter xni only part of the energy was represented by standing 
waves, the rest energy being taken care of by the external wave function. 
But here the whole curvature of the Einstein universe has been replaced by 

ETP 17 



258 Physical Applications [14-1 

wave functions i/f, < in a flat space-time. Thus the density p = 3P in (14- 144) 
is the whole of the density. 
Setting p = 3P in (14-12) we obtain 

A = f#- 2 , 87rKp = fJ&- 2 . (14-151) 

Let us now revert to the usual standpoint, and regard JB as a radius of 
curvature. The waves are then abolished; they were a representation of 
the projection factor j3 which is no longer used. Since there are no standing 
waves, P = 0. Inserting P = in (14-12) we obtain 

A' = J2- 2 , STTKP' = 2JR- 2 , (14-152) 

accents being used to distinguish this from the previous reckoning. We have 

. p'=$P- (14-153) 

The change of reckoning from /> to p' depends on the change of the 
arbitrary constant in the energy tensor from A to A'. We have seen ( 13-1) 
that this signifies a change of zero point from which energy and pressure 
are reckoned. When we represent the curvature by waves we introduce a 
pressure, uniform throughout the universe, which (according to our usual 
outlook) is fictitious. The change of zero point gets rid of this pressure, and 
at the same time changes p to p. 

The pressure (in ordinary reckoning) is not necessarily zero in an Einstein 
universe. Static universes with non-vanishing pressure are obtained by 
taking A slightly greater than J?- 2 . But we are here treating a radiationless 
universe, which is accordingly at zero temperature, so that the pressure 

is zero.f 

By (14-144) and (14-163), />' = 3/72^7. The total mass or energy of the 
Einstein universe in ordinary reckoning is therefore 

M'=p'V=*IR*m. (14-161) 

There is further the well-known relation, obtained from (14* 152) and (14- 143), 



Thus if M' or R is given we can determine the mass-constant w, which 
occurs in the definition of the energy operator. 

If there are two reckonings of m as there are of p and M 9 viz. m 1 = fw, 
(14-161) can also be written ^ _ 4/^2^ (14-162) 

The origin of the factor f is that, since we are treating static conditions, 
the gauge transformation is applied in three dimensions only. The gauge 
transformation in 13-7, which gave the most straightforward formal 

f According to (13*27) the pressure at zero temperature is not strictly zero. For the 
Einstein universe this residual (degeneracy) pressure is found to be of order 10~ 26 p, and could 
be neglected. It is, however, the formula (13*27) which should be corrected by this small 
amount (to allow for the cosmical curvature); and the degeneracy pressure in the Einstein 
universe is strictly zero. 



14-2] The Cosmical Problem 259 

connection between waves and curvature, was in four dimensions. Con- 
sequently the four-dimensional volume element is changed in the ratio j8 3 
in this section, whereas it was changed in the ratio jS 4 in 13'7. Another 
example of this factor occurs in the two modes of reckoning magnetic energy 
(12-85) and (12-86), according as the field is strong enough to render it 
necessary to treat the problem in four dimensions or weak enough to be 
treated as a perturbing effect in a three-dimensional problem. 

14-2. Analysis into Particles. 

In the foregoing calculation we have treated the uniform density p of an 
Einstein universe as though it were the probability distribution of a single 
particle of mass-constant m. We shall now suppose that the space of radius 
B is occupied by N' elementary scalar particles. By the exclusion principle 
these will be represented by orthogonal wave functions, each particle having 
a separate wave function. To obtain the ground state of the distribution 
we must select the wave functions of lowest energy. 

The wave functions can be classified similarly to those of an atom, except 
that we are not troubled by the duplication of states due to opposite direc- 
tions of spin. (That is taken care of automatically when the N r scalar 
particles are replaced by 4JV' elementary vector particles.) In the atom the 
waves are concentrated into small volume by a controlling Coulomb field. 
Here the gauge factor /? plays the part of a concentrating force; or, what 
comes to the same thing, the waves spread to the natural limit imposed by 
the finitude of space. In the stereographic projection of the uniform sphere 
we have a concentration of density towards the origin and a thinning out at 
great distances, just as in the atom; we may look on it as an illusion of 
projection, but mathematically the effect is the same as if the distribution 
were being held together by a central force. 

The lowest state (K state) is the projection of a uniform^ spherical dis- 
tribution; and the investigation of 14- 1 is directly applicable to it. We need 
not treat in detail the successively higher states, which will be projections of 
spherical harmonics in four dimensions. We proceed at once to the highest 
of the N 1 occupied states, which we shall call the limit state. We denote the 
energy of the limit state by w 2 . For the actual universe the limit state 
corresponds to very high quantum number (about 10 26 ), so that the energy 
levels have there become practically continuous. For the ordinary problems 
of physics (excluding cosmical problems) there is an inexhaustible supply of 
particles with energies practically equal to the limit energy m 2 . 

The mean energy of a particle is 

m = fw 2 . (14-21) 

This is a general formula applying to all systems of orthogonal functions in 
three dimensions. For the rectangular waves, treated in the box problem, 

17-2 



260 Physical Applications [14*2 

it is contained in the formula S 2 = fr 2 (13-26); it there follows from the fact 
that the elementary wave functions form a lattice of uniform density in 
to-space. It is well known that the density of the lattice remains invariant 
when other orthogonal functions are used instead of Fourier functions, and 
when the controlling field of potential is varied; or, as it is usually expressed, 
each eigenfunction occupies a cell of phase space of volume A 3 . The only 
addition we need make here is that, since p = 3P for standing waves, the 
ratio between the mean contribution and the limit contribution to the 
pressure applies also to the energy density. 

In treating the energy levels in an atom, recourse is had to a device known 
as the selj '-consistent field. The electrons are put one by one, not into the field 
of the nucleus alone, but into the field provided by the averaged distribution 
of the whole system. We use the same principle in analysing the universe 
into particles. The whole distribution determines the radius R of the space, 
into which we put the particles one by one; and the controlling field j8 9 
which depends on R, is likewise determined by the complete distribution. 

Let us glance at the alternative of building up the Einstein universe 
synthetically. The first particle takes up a distribution of self-equilibrium 
a miniature Einstein universe consisting of one particle. We can calculate 
its energy by (14-162), since in this case Jf' = m'. But the calculation is of 
little value, since the K state in the completed system will have an altogether 
different energy. Each particle that is added, not only contributes its own 
energy, but modifies the energies of all the preceding particles. 

By the use of the self-consistent field the energies of the particles are 
defined in such a way that they are precisely additive. This is important in 
connection with the question, whether the whole density of the distribution 
is represented by X ift k <f> k or by (S $ k ) (S </> k ), $ k and </> k being elementary wave 
functions. In ordinary applications, such as the problem of the rectangular 
block, it does not matter; because, owing to the orthogonality of the wave 
functions, the cross terms yield zero integrals over the small volume that is 
being considered. But when the waves extend over the whole universe, the 
vanishing of the integral over the whole universe is not a sufficient reason for 
neglecting the cross terms locally. When we use the self-consistent field the 
density is fc < fc . For the wave functions, and the energy derived from them, 
are calculated on the supposition that all the particles are present and 
producing the field j8. The cross-energy with all the other particles, as well 
as the self-energy of the kth particle, is already incorporated in $ k <f> k \ and 
we do not require additional terms ift k <f> k > to represent it. 

This is the same outlook as in 12*6, where the mutual energy invariant 
of two systems is replaced by self-energy invariants. In fact the energy 
attributed to a particle must necessarily originate as a mutual energy of some 
kind. It is only when the distribution of the other particles differs from the 



14-3] The Cosmical Probkm 261 

standard distribution used in calculating the supposed self-energy, that the 
mutual energy terms have to be recognised explicitly .f 

A precaution is necessary is using the self-consistent field, if we are to 
avoid counting the energy twice over. In a field of fixed potential < the 
energy of a particle is reckoned as m^\ but in a field produced by other 
particles the energy is reckoned as Jw^, the other half being allotted to the 
particles producing the field. 

We may summarise the progress of ideas as follows. The mechanical 
properties of macroscopic matter forming an Einstein universe are usually 
represented by curvature of space-time; but we can, if we prefer, analyse 
it into particles and embody the mechanical properties in wave functions. 
Energy and pressure represented by wave functions must be omitted from 
the curvature (11-7); thus the complete wave representation consists of 
wave functions in flat space-time. We therefore consider an alternative 
representation in flat space-time obtained by stereographic projection of 
the distribution. In this representation there is a concentration towards the 
origin, as though a controlling force prevented the matter from spreading 
away to infinity. We analyse this density distribution into orthogonal wave 
functions, representing particles which obey the exclusion principle. All the 
particles must be fully present in the distribution, up to a limiting energy 
determined by the number of particles; otherwise there would be a conflict 
between the relativity conditions and the wave mechanical conditions for 
a static or ground state. Conceptually each wave function or state could be 
occupied or not; but actually the decision that the states shall all be occupied 
was taken at the beginning when we chose a radiationless static universe 
for projection. Thus the disturbance of a particle by the particles occupying 
other states has been taken into account from the beginning. The test of 
consistency is that the sum of the densities corresponding to each state shall 
reproduce the density originally postulated. 

14-3. Threshold Energy. 

In elementary quantum theory the system under discussion is always treated 
as an independent addition to the rest of the universe. If the quantum phy- 
sicist ever remembers that there is a "rest of the universe ", he treats it as an 
ideal background which will not interfere with his system. He does not 
picture it as consisting of N' other particles competing with his own particles 
for the states of lowest energy; nor does he contemplate the possibility of his 
own particles dropping into vacant levels in the background. The back- 
ground is treated as impermeable. 

This is equivalent to assuming that the "rest of the universe" is in the 

t Thus perturbation theory will involve (2ty ft ) (S# A ) since it deals with deviations from the 
distribution which furnishes the adopted self -consistent field. 



262 Physical Applications [14-3 

ground state; so that there are no excited particles to drop into the vacant 
levels in the added system, and no vacancies for the added particles to fall 
into. 

When another (scalar) particle is added to the system of N' particles in 
the ground state, the lowest vacant energy level is at the limit energy m 2 . 
The added particle must be endowed with an energy w 2 in addition to any 
transferable energy it may possess. Thus w 2 , which is the limit energy of the 
background particles, is the threshold energy of the added system. There is 
thus no interference between the added particles and the rest of the universe. 
The energy levels in the added system begin at w 2 ; the particles forming the 
rest of the universe are all placed below this threshold level. 

The threshold energy w 2 is the rest energy or proper mass of an added particle. 
This applies to scalar particles; it is modified in accordance with the theory 
of Chapter xn when we substitute charged particles. The added particle 
may be excited above the threshold level, and so possess additional (kinetic) 
energy. The transferable energy of a particle is really its energy of excitation 
in the universe-atom; though it is, of course, not usually regarded that way. 

Although the elementary equations of quantum theory postulate that the 
particles forming the rest of the universe are in the ground state, that does 
not mean that the theory is only applicable if the universe is an Einstein 
universe. Of necessity an elementary equation refers to idealised con- 
ditions. The ground state is the fixed standard background; and any devia- 
tion from the ground state must be explicitly described as an addition to 
the fixed background, and as such taken account of as part of the added 
system. Usually these additions are called "fields". 

For example, let the added system consist of particles in thermodynamical 
equilibrium at temperature T. Clearly there must be statistical equilibrium 
between the added particles and the background particles, so that the back- 
ground particles will be excited. But the excited background is not treated 
as such. It is described as a fixed impermeable background plus a field of 
radiation. Exchanges of energy between the added particles and the 
excited background are described as exchanges between the added particles 
and a field of radiation. 

Radiation is the name under which vacancies in the sub-threshold levels 
are taken into account in current theory. I can see no possible doubt about 
this identification. For in problems of this type radiation is the only entity, 
besides the particles, that is mentioned as an addition to the fixed back- 
ground. So that, if the vacancies are not taken into account as radiation, 
they are not taken into account at all which would clearly be a gross error 
rendering agreement with observation impossible. 

I think that this is the natural approach to a quantum-relativistic theory 
of radiation. But as radiation is rather apart from the main subject of these 



14-3] The Cosmical Probkm 263 

investigations, I have not found time to develop the theory. I do not suppose 
that progress will be easy; for it is a long step from radiation in an Einstein 
universe to radiation in the concentrated systems where it is studied in 
practice. That radiation is the positive aspect of the vacancies caused by 
excitation of the unspecified (scalar) particles seems to me indubitable; but 
what I now add is an unchecked first impression. 

Presumably a single vacant level constitutes a photon. The energy of the 
photon is that which is released when the photon is abolished by a limit 
particle falling into the vacancy, i.e. the depth of the vacant level below the 
limit energy. Since the. energy levels are practically continuous near w 2 , a 
photon can have practically any energy up to w 2 . I doubt if ra 2 is a genuine 
upper limit; further investigation would be required to ascertain whether it 
still applies in less idealised conditions. 

In non-equilibrium conditions the added particles may be localised as 
wave packets-, similarly the vacancies may be localised as voave pockets. The 
wave pocket seems to be nothing more than the partial localisation of a 
photon, manifested, for example, in the observation of individual X-ray 
effects in an expansion chamber. 

The particles which should have occupied the vacant levels will exist at 
or above the limit level. (A vacancy, in the sense of annihilation of a particle, 
is not a photon but an impossibility.) If a photon forming part of a field of 
radiation is absorbed by an atom, we can picture it as continuing its exist- 
ence in the same form inside the atom, namely, as a vacancy at one of the 
lower levels in the atom. In such exchanges the energy of the photon, 
measured by the depth of the vacancy below the limit up to which the levels 
are occupied, does not remain constant, since the ejected particle may carry 
off some of the energy. 

Leaving the subject of radiation,f we return to the added particles. The 
added system is formed by specifying certain particles either by wave 
functions or macroscopically. The particles to be specified must always be 
taken from the uppermost level, so as to leave no hole in the background. 
Since the levels near m 2 are exceedingly close together, there is for ordinary 
purposes a practically unlimited supply of particles of rest mass m a ; and the 
specified particles will have this proper mass. But when we extend the 
formulae to cosmical systems, we must ultimately come to the deeper 
strata. Finally, when we apply our calculations to the whole Einstein 
universe, and specify the whole of the N 1 particles (macroscopically), the 
total rest energy instead of being N'm 2 is 

M=N'ffi=%N'mt (14-31) 

by (14-21). 

t For a deduction that photons (as here identified) obey Einstein-Bose statistics, see 
16-3. 



264 Physical Applications [14-3 

The deficit f N'm% is the (negative) gravitational potential energy. 

Gravitational energy is therefore the exclusion effect treated in 14-2, 
but viewed from the highest level instead of from the zero level. Remem- 
bering that the unspecified particles have to form an impermeable system, 
each successive particle that is transferred to the specified system must be 
taken from a slightly deeper level. The rest energy of n particles is slightly 
less than n times the rest energy of one particle. When we come across this 
phenomenon observationally we attribute it to gravitational energy. We 
have here considered only the simple case of an Einstein universe, and have 
not entered into the complications due to concentrating the specified 
particles. That is scarcely necessary; for we are not putting forward a new 
"explanation" of gravitational energy. The relativity representation of a 
gravitational field as curvature of space-time was the starting point of the 
whole investigation. What we have now reached is the same phenomenon 
viewed from the topsy-turvy outlook of elementary quantum theory. 

Exclusion energy, interchange energy, potential energy are different ways 
of regarding the same thing. Up to the present we have been concerned only 
with gravitational potential energy; but in Chapter xv we shall find that 
electrical potential energy has the same origin. 

There is reason to think that our expanding universe is rather far removed 
from the Einstein state and that a de Sitter universe would be a better 
approximation. It may be suggested that this, whilst not invalidating the 
use of an Einstein universe as a standard background, may cause it to be 
inconvenient in practice. But the local conditions of our experiments always 
differ much more widely from those of an Einstein or a de Sitter universe 
than these do from one another. The density in an Einstein universe is 
3-32. 10~ 27 gm. cm.~ 3 The use of an Einstein universe as the standard con- 
dition, to which the exact equations refer, means that gravitation corre- 
sponding to this density has been allowed for in the equations. The actual 
field in any practical problem is always very much greater, and it is unim- 
portant whether the standard field is deducted or not. 

14-4. Positrons and Negatrons. 

The scalar wave functions, which we have been considering, can each be 
analysed into four vector wave functions representing protons and electrons. 
We may therefore have at any level a vacancy due to the absence of a proton 
or electron, instead of the absence of the whole scalar particle. We have seen 
that the absence of a scalar particle is represented as the addition of a 
photon. According to Dirac the absence of an electron is represented as the 
addition of a positron. 

I think that the present investigation throws new light on the meaning 
of the vast number of occupied negative energy levels, which Dirac postu- 



14-4] The Cosmical Problem 265 

lated in his theory of the positron. The levels are described as negative, 
because energy is now reckoned from the threshold energy as zero.f This is 
in accordance with the outlook of elementary quantum theory described 
in 14*3; the systems which it treats are additions to the fixed background, 
and the zero of energy is therefore transferred to the threshold level at which 
the additions start. Of these negative energy states, Dirac saysf 

These assumptions require there to be a distribution of electrons of infinite 
density everywhere in the world. A perfect vacuum is a region where all the 
states of positive energy are unoccupied and all those of negative energy are 

occupied The infinite distribution of negative -energy electrons doea not 

contribute to the electric field. 

We have reached an altogether different view of the way in which the 
negative energy levels are filled. They are occupied by the large amount of 
matter of the universe not specifically mentioned in the equations which 
Dirac was developing and applying. This matter is ignored, i.e. treated as a 
fixed background, in the idealised equations; and such treatment auto- 
matically relegates it to the levels below the threshold which it must fill 
completely. Of course, in the actual universe this is far from true; but the 
idealised equations are a far from complete representation of the conditions, 
and must be supplemented by terms representing fields of radiation, etc. 
The chief points on which we go beyond Dirac's theory of the positron are : 

(1) Our negative energy levels are occupied by protons and electrons 
equally. Dirac supposes them to be occupied by electrons only, with the 
result that there is an infinite negative charge to be suppressed in an 
arbitrary way. 

(2) Our theory places protons and electrons on the same footing, and 
therefore definitely predicts the existence of negatrons or negative protons, 
unless there is some unforeseen limit ( < 2m p ) to the amount of energy that 
can be expended in a single process of excitation. 

(3) The number of negative energy levels is not infinite; it is known 
precisely ( 14-7, 16-8). 

When an electron annihilates a positron, the sum of their energies 
( ^ 2rn e ) is emitted as radiation; In Dirac 's theory it is assumed that this 
is equal to the difference of energy level, as though the falling of the 
electron into the vacant level were the same kind of transition as that 
which gives a spectral line of an atom. But in the annihilation of an 
electron and positron in free space, two photons are emitted. The process 

t This, however, is not the negative energy to which Dirac refers. A positron has rest 
energy m e , and he therefore supposes that the particle whose absence it represents would 
have had rest energy m e . As explained at the end of this section, we do not accept 
this conclusion. 

J Quantum Mechanics, 2nd ed., p. 271. 



266 Physical Applications [14-4 

is therefore not comparable with an ordinary transition in which the whole 
energy is emitted as one photon. 

In our theory the depth of the vacant level below the zero level gives 
only the excess of the energy of the positron above its rest energy m e . 
A vacancy just below the limit level corresponds to a positron almost at 
rest. We have seen ( 6-7) that owing to the idempotency of the stream 
vector of an elementary particle, the vector J representing charge, current, 
etc., is often confused with the vector J 2 representing energy, momentum, 
etc. The absence of an electron whose stream vector would have been J 
is equivalent to the presence of a positron with stream vector J\ the 
reversal of the charge and current is thus duly indicated. But the energy 
and momentum of the electron and positron are J 2 and ( 7) 2 , so that 
they are the same; in particular, the quarterspur which represents the 
proper mass is the same. Our later developments of the theory of the 
origin of mass have thrown some more light on this point. We recall that 
energy is furnished to a system by specifying it (11-6), and that the 
energies m p9 m e are additions made at one stage of the specification, 
viz. when a neutral particle is specified as a particle with definite charge 
and spin. So far as this addition is concerned, it makes no difference 
whether the original neutral particle had a positive or a negative existence, 

14*5. Masses of the Added Particles. 

It is useful to distinguish two systems of application of wave mechanics: 

A. Cosmical System. This treats the specified and unspecified particles 
as one great system of N' particles. Owing to the exclusion principle the 
particles occupy various energy levels; and in the ground state their energies 
extend up to a limit m 2 . 

B. Local System. The zero of energy is moved up to the level m 2 , and the 
local system is an addition above that level. On the negative side of the new 
zero there remains the completely filled set of levels (now negative) which is 
never disturbed. This is variously regarded as a comparison fluid, an im- 
permeable background, a pure inertial field, or a static spherical space, on 
or in which to erect system B. 

A disturbance of system A from its ground state necessarily involves 
defect below the limit level as well as excess above it. But nevertheless in 
system B the levels are always assumed to be completely filled; and the 
defect as well as the excess is represented as an addition above the new zero. 
Suitable entities, such as radiation, positrons, electromagnetic potential, 
are introduced to represent the defects when considered as additions. In 
Newtonian theory an irregular gravitational field is also regarded as an 
addition to the fixed background. If we follow Einstein's theory, it is not an 
addition to but an abandonment of the fixed background. A space of variable, 



14-5] The Cosmical Problem 267 

and in general non-static, metric is substituted for the uniform static 
background. System B is then inapplicable and it is necessary to revert to 
system A. 

We shall now consider more closely how the electrons and protons of the 
local system are created as an addition to the fixed background. We take 
a neutral particle (a quarter of a scalar particle of system A) at the limit 
energy, and "specify" it. That is to say, we assign to it wave functions 
giving its probability distribution as modified by incorporating observa- 
tional information as to its charge, spin, momentum, position, etc. As a 
neutral particle it was equally likely to be a proton or an electron; we now 
specify it definitely as an electron, say. This involves the use of vector wave 
functions. The vector wave functions also make the particle mobile in the 
ordinary sense, i.e. characterised by a stream vector to which Lorentz 
transformations are applicable. The scalar wave functions can only express 
mobility of the type represented by standing waves. 

Let us now regard spherical space as the frame, the original probability 
distribution of the neutral particle as the partial comparison fluid, and the 
specified distribution as the object particle. The observable relations are 
contained in the double wave vectors T, X of the combined system. It is 
in these that the observational information is directly incorporated; for 
example, our observations create wave packets in *F, X. Current wave 
mechanics replaces these by simple wave vectors ^, x for the object particle 
and scalars ^, o> for the comparison fluid. As shown in Chapter xn this 

imposes the relation a ,,,*,, v 

* 10m 2 -136ww + m 2 =0 (14-51) 

between the mass m of a proton or electron in the local system and the mass 
m of the neutral particle. 

Remembering that the addition of two systems is represented by multi- 
plication of their wave functions, we notice that whereas V describes a 
modification of the distribution, $ describes what has been added by the 
modification. The reduction from T to is really a casting out of the original 
background particle. If we define an electron or proton to be the entity re- 
presented by ^r, it is a pure addition to the background. Accordingly the 
simple wave vectors are the appropriate representation in system 5. This 
may be more easily seen if we use energy invariants as in 1 2- 6. By specifying 
with double wave functions a neutral particle of system A which had an 
energy invariant w 2 , we obtain a particle, also in system A, with energy 
invariant 136wm . This is depicted in system B as having an energy in- 
variant 10m 2 . By (14-51) there is left an amount m 2 to provide for the 
original particle which remains in the background. Thus the procedure of 
specification, besides providing a particle in system , stops up the hole in 
the background. 



268 Physical Applications [14-5 

Our result that the rest mass of a particle is to be identified with the 
threshold energy w 2 applies to scalar particles. The mass becomes multiplied 
by i(w p + ra c )/w = 136/20 when the scalar particles are replaced, as they 
eventually must be, by protons and electrons. We may regard the factor 
136/20 as a transformation of the scale of reckoning in passing from system 
A to system B. 

We can now see the complete connection between the masses of the 
elementary particles in quantum theory and the conception of mass as 
curvature in relativity theory. In quantum theory piass corresponds to the 
periodicity of the waves. The only direct connection of periodicity and 
curvature is the periodicity which arises from "going round the world", 
the corresponding wave length being 2?rJ?. The corresponding mass in c.o.s. 
units is A/27TJSc. At first sight this seems far too small to be important. But 
we have to remember that there can be only one scalar particle to which this 
applies; and the exclusion principle forces succeeding particles to have 
periodicities corresponding to the higher harmonics. The result is that an 
average particle has a mass comparable with that of an electron or proton. 
But there is still the factor 136/20 to be applied before we reach the masses 
ordinarily recognised. It is not necessary to go over again the explanation 
of its occurrence in Chapter xu. We may note, however, that since the rest 
mass is not an intrinsic attribute of the particle but represents the energy 
of the particle in an assembly in statistical equilibrium, it depends on the 
number of degrees of freedom which share in the equipartition of energy. 
The factor is really a compensation for adopting (in current theory) a 
simplified representation in which the recognised number of degrees of 
freedom is greatly reduced by the substitution of simple for double wave 
functions. 

14-6. The Standard Mass m . 

It is perhaps necessary to remind ourselves that we are not putting forward 
a new theory of natural phenomena; we are comparing two mathematical 
methods of treating the same system. Primarily we are treating an Einstein 
universe, i.e. a uniform static distribution of matter extending indefinitely; 
but in order to compare our scale of measurement with that used in quantum 
theory we insert in it a microscopic "added system". From the ordinary 
point of view (system B) the added system is the centre of attention; the 
equations used to describe it in elementary quantum theory take it for 
granted that its surroundings are uniform and static, as here supposed. The 
surroundings are normally represented by a metrical field g^, which in the 
uniform conditions postulated corresponds to a spherical space of radius It. 
The value of B is involved because, by the macroscopic theory, the material 
standard of length used in actual experiments on the added system takes up 



14-6] The Cosmical Problem 269 

an extension which is in a definite ratio to the radius of curvature of the 
corresponding three-dimensional section of space-time.f Thus we have to 
treat an added system represented by waves in surroundings represented by 
curvature. We now introduce a mathematical transformation of the pro- 
blem, which represents them both together by waves (system A). We first 
represent the uniform surroundings, or Einstein universe, by waves. The 
added system is then formed primarily by a modification of these waves 
a specification by double wave functions. But the modification can also be 
represented by the original waves together with additional waves; and the 
additional waves form the usual description of the added system. 

It is important to realise that the physical system studied in this 
chapter is precisely the physical system which a quantum physicist treats 
at the beginning of his subject. We are treating an Einstein universe; 
that is all except that we forgot to mention that one of its particles must be 
excited (one will be enough) in order that the natural constants h, m e , etc. 
may appear in the problem. The quantum physicist takes as his most 
elementary problem one particle in free space ; that is all except that he 
forgot to mention that there are some 10 79 other particles present in an 
equilibrium state. (For he believes that his formulae can be experimentally 
verified without destroying all the particles in the universe except one.) 
Our problems are identical; but our respective forms of absent-mindedness 
show that we view them from a different outlook. The two outlooks 
correspond to system A and system B. 

We must notice one point in the transformation which will be important 
in the numerical calculations about to be made. The limit state of system A 
becomes the K state of system B. We define the K state as that in which a 
particle has uniform probability distribution over spherical space. The 
threshold energy m z is that of a particle definitely at rest in system B and 
therefore having entirely uncertain position. The added particles in system 
B do not exclude one another, except for the slight negative exclusion effect 
represented by gravitational potential energy; thus, except in so far as the 
position is specified, any reasonable number of them can occupy the K state 
of uniform distribution. This change of K state occurs when we pass from 
the universe with p = 3P to the universe with P = the change which 
introduces the factor (14-153). In abolishing the pressure, the energy 
previously represented by standing waves is replaced by curvature. We 
must not duplicate the representation by waves and curvature. Energy up 
to the threshold energy being represented by curvature, only the excess 
remains to be represented by waves. Thus in system B a particle at rest is 
represented by a uniform distribution of probability, instead of by standing 
waves corresponding to a spherical harmonic of high order. 
t Maihematioal Theory of Relativity, 66. 



270 Physical Applications [14-6 

The connecting link between systems A and B is the mass ra of the 
neutral particle. In system B it has been defined as the mass belonging to a 
simple (progressive) wave function which possesses only an algebraic phase; 
equation (14-51) was derived as the necessary relation between the mass w 
of such a wave function and the mass m ( = m p or m e ) of a vector wave func- 
tion both referred to the geometrical frame. In system A our elementary 
scalar wave functions have mass w 2 ; but the waves are of a somewhat 
different character, being standing waves in three dimensions instead of 
progressive waves in one dimension. It would not have been possible to 
amalgamate progressive waves in three dimensions without introducing 
matrix phases. 

We already know that the scalar particle mi corresponds to four neutral 
particles ra , which on specification yield two protons and two electrons. 
There is an apparent discrepancy between the four particles and the three 
dimensions of the standing waves. This is rectified by the factor f found in 
(14-153). It is true that W = |w 2 ; but in ordinary reckoning w a is replaced 
1* <-*.. "that ^^ (14<61) 

It is convenient to get rid of the factor f by introducing an intermediate 
system A', which is the same as system A except that all masses are multi- 
plied by $. 

Similarly we introduce an intermediate system B' consisting of neutral 
particles with one-phase wave functions. The multiplication of the mass by 
136/20 then occurs in passing from system B' to system JB. 

We have then four systems defined as follows. System A is the direct result 
of the analysis of the curvature into elementary scalar wave functions 
constituting standing waves in three dimensions. In system A' we convert 
the Einstein universe with p = 3P into an Einstein universe with P=0, 
thereby increasing all energies in the ratio f ; this increase is the same as if 
the energy had been calculated for standing waves in four dimensions instead 
of three, and we shall regard it in that way. In system B' some of the limit 
particles of system A' with energy ra 2 ' are taken as the basis of an added 
system; the scalar particle with (effectively) four degrees of freedom is for 
this purpose divided into four neutral particles with one-phase wave 
functions and energy w = |w 2 '. In system B the ordinary vector wave 
functions are introduced and the total mass becomes multiplied by 136/20. 

In system JB' the coefficient m in the energy operator m-*8*l8xp8x a can 
be identified with m . This follows from the elementary formula for the 
plane waves of a neutral particle 



m = i d/ds = mo" 1 
This must also apply to system A 9 . But in system A the unit of mass is 



14-6] The Cosmical PrdbUm 271 

altered so that m=|m . Thus (14*144), which refers to system A, becomes 

P-l/JIXF, p-S/JZXF. (14-621) 

Hence the corresponding total mass in system A' is 

JT-fJlf-ipF-4/lZX, (14-622) 

which is the more explicit form of (14-162). This is the expression for the 
energy of a scalar particle forming a uniform distribution in spherical space 
of radius jR, i.e. it is the energy of the K state. We shall therefore change the 
notation M' to m K . It is convenient at this stage to insert the factor A/TT 
required to reduce masses to c.o.s. units.f We therefore write (14-622) as 



* m Q R 2 \7rf 

As we shall not further use system A, we shall omit the accents previously 
used to distinguish quantities referred to system A'. 

Consider first a universe which consists of one scalar particle. There can 
be no absolute comparison of standards in different universes; we shall 
therefore arbitrarily define the corresponding units to be such that m and 
R are the same as in the actual universe. The masses m 2 , m p , m e which are 
numerically connected with m will also be the same. But the natural con- 
stants K, h may be different. We denote their values in the one-particle 
universe by /c x , h l . 

In the one-particle universe, the K energy, the limit energy, and the total 
energy coincide, so that m K =m 2 = M. Hence, by (14-63), 

(14-641) 



so that Wo^/TrJZ. (14-642) 

Returning to the actual universe, let 

P = ? S N'. 

Then by (14-31) w a = M/p. That is to say, the wave function representing a 
single scalar particle uniformly distributed over the sphere J will have only 
1/pth of the energy required to produce the curvature 1/-B. In (14-641) a 
single particle produced the curvature 1/12, and the coefficient mr l of the 
energy operator was then m^ 1 (At/Tr) 2 ; we must therefore take the co- 
efficient to be 1/pth of this, namely m Q " 1 (A/w) 2 , where 

tf^p-ihi*. (14-66) 

From (14-642) and (14-65) 

(14-66) 



f We still omit the constant c. For the factor h/ir, see 9-6. 

j As explained earlier in this section the threshold energy w a corresponds to uniform 
probability distribution over the sphere in system B. System B corresponds to the ordinary 
equations of elementary quantum theory which define the constant h. 



272 Physical Applications [14-6 

We may note also that ic 1 =^/c, since the well-known condition for an 
Einstein universe, applied firstly to the one-particle and secondly to the 
tf'-particle universe, gives Kim ^^ R = Kpm ^ (14-67) 

Thus K/h* is independent of N'. 

14*7. Numerical Solution. 

Since each scalar particle corresponds to four elementary particles, the 
total number of protons and electrons is N = N f . Inserting the constant c 
to reduce to C.G.S. units, (14-66) becomes 

m Q = hV%N/27rRc. (14-71) 

We have also, by the ordinary relativity formula for an Einstein universe, 

\<nE = >cJ//c 2 = \KN (m p + m e )/c* (14-72) 

together with m p + m e = ( 136/10) m (14-73) 

by (14-51). 

From (14-71) and (14-72) 

N 7TC 2 



whence V^= _ = _ - _ Q4-74) 

* ~2 K m Q (m p + m e ) 10 2 K (m p + w c ) 2 v } 

by (14-73). We shall find in Chapter xv that Ac/27re 2 = 137; so that the result 
can also be written as 



10 ic( 

Formula (14-75) is the most suitable for an accurate determination of N 
from observation. As suggested by W. N . Bond the so-called ' ' observational ' ' 
values of e/m e are (from the point of view of wave mechanics) erroneous by 
a factor ^|f ; but the "observational" values of m p /m e are also in error by 
the same factor, so that the observational values for ejm p can be accepted as 
correct. I adopt the values given by W. N. Bondf from recent deter- 
minations: 

e//w e c= 1-7574. 10 7 (uncorrected); 1-7703. 10 7 (corrected), 
so that e/m p c= 1-7703. 10 7 ~ 1847-6 = 9582. 

For K the generally accepted value appears to be 6-664.10~ 8 , but it is 
uncertain to 1 part in 1000. Inserting these values in (14-75), we obtain 

JV P = 3-1454.10 79 . 

Following a suggestion by R. Fiirth that the core of the large number N 
is 2 256 , we can express the number %N of electrons or protons as 

#= 135-82. 2 256 . 
t Nature, 135, 825 (1935). 



14-8] The Cosmical Problem 273 

Since N is an integer by definition, no irrational factors can enter into its 
composition; and the possible hypotheses as to its constitution are very 
limited. In view of the close association of the numbers 136 and 256 in the 
theory of the double wave function, our result makes it probable that the 
exact value is JV = 136. 2 256 . In Chapter xvi we shall obtain this number 
independently from theory. 

Conversely, assuming JJV = 136 . 2 266 , and using the observational values of 
e/m p and c, we can calculate K. The result obtained by Bondf is 

* = (6-659 -0012). 1C- 8 . 

If the ratio of the electrical to the gravitational force between a proton and 
electron (calculated according to classical theory) is denoted by F, we have 

F== & = 1362 g2 

~ *cw, n w ~ 10 *c(w 



| QO _ 

Hence ( 14-75) gives F = - VN. ( 14-76) 



14-8. Alternative Treatment. 

If the scalar wave functions ^r, <f> contain a time factor e ikt , the particle 
density a contains a time factor e 2?7f/ . The volume is therefore expanding as 
e -2iM } an( j fa e linear scale as e~$ ikt . We can therefore connect the represent- 
ation of mass by waves with the theory of the expanding universe. 

Consider an expanding spherical universe whose radius at time t is 
R = K Q ef. The line element is J . 



x 9 y, z being the stereographic coordinates at time t. The pressure and 
density are found to be 



To represent the above imaginary expansion of linear scale, we must take 

(14-825) 



so that (14-82) becomes 

^P=-^ 2 -* 2 

V ' 



), J 
which is a generalisation of (14*12). 

t Nature, 137, 317 (1936). 

% Tolman, Relativity, Thermodynamics and Cosmology, equations (150-2), (150-7), (150-8). 

ETP 18 



274 Physical Applications [14*8 

As in 14-1, we replace the curvature by standing waves in flat space, so 
that p = 3P. Inserting this condition in (14-83) we have 

f& 2 + A = f JB-*= Sine/), (14-84) 

which is a generalisation of (14-151). 

The meaning of (14-84) is that, by adopting a slightly lower value of the 
arbitrary constant A, we can reserve a portion of Sirup to be represented by 
wave functions with a time factor e ikt . The reservation of a portion of the 
whole energy tensor or matrix for representation by waves has been dis- 
cussed in 11-7. Let l/pth of the energy tensor be thus reserved, so that 

|Jfc 2 = 3/2jpJZ 2 . (14-85) 

There are two ways of viewing (14-84). The macroscopic view is that 

A = SlTKip, %k 2 = &irK 2 p, 

so that the energy of the wave is provided by a general weakening of the 
gravitation constant involving a decrease of the gravitational mass of every 
particle in the distribution. But in that case the factor e iki belongs to a 
collective wave function of the whole particle density, and not to an in- 
dividual particle. The alternative view is that 

A = SlTKp! , ^fc 2 = &7TKp2 . 

In particular, if p = f N', the reserved density p 2 amounts to just one scalar 
particle in the K state. This corresponds to the elementary conception of 
a single particle represented by waves, with an impermeable background 
represented macroscopically. 

The mass corresponding to a time factor e ikt is 



In the case of a collective wave function, every particle has this amount of 
mass represented by waves; for its own particle density (equal to the 
product of its individual wave functions) changes with the expansion of R 
in the same proportion as the whole particle density. The total mass repre- 
sented by the waves is therefore hpk/7T. f This mass is now replaced by a single 
reserved particle whose individual wave function must accordingly be given 
a time factor e ipki . In ordinary reckoning (system A') the mass of the 
particle is multiplied by f , and becomes 



Hence, by (14-85), m ' 2= * }' (14-86) 

This result must, however, be divided by 2, because the energy density p 2 
here attributed to the particle is a mutual energy of the particle and the rest 

t That is to say, if the individual wave functions are of index 1, the collective wave 
function is of index l/p. 



14-8] The Cosmical Problem 275 

of the system; so that if we sum it for all the particles we obtain twice the 
total energy. It will be seen from (14*84) that if we reserve in succession the 
p equal amounts of density /t> 2 , the unreserved portion of -R~ 2 (which is left 
to be represented in system A' as curvature) diminishes by equal steps, so 
that m' 2 decreases regularly from (14-86) to zero. The corrected result 
(giving the energy reckoned according to the method of the self-consistent 
field) is therefore 

(14 ' 87) 



The mass m f here found is to be identified with m . It is perhaps rather 
difficult to see that it is ra rather than w a . But it is to be remembered that 
the mass constant both for the scalar particle and the neutral particle is ra ; 
and the fourfold energy of the former comes from its additional degrees of 
freedom. We are here treating the progressive waves in one dimension t\ and 
(14-87) applies to one particle with one degree of freedom. If we represented 
the four neutral particles by separate wave functions, each of them would 
have the time factor e ipki and the corresponding energy; just as, when earlier 
we divided the collective wave function into p individual wave functions, 
each of them had the energy corresponding to the time factor. 

The result (14-87) is accordingly an alternative derivation of (14-66); and 
the solution in 14-7 then follows. 

If we are not seeking the exact result, it is comparatively easy to obtain 
a solution of the cosmical problem sufficient to confirm the order of magni- 
tude of the recession of the spiral nebulae. 

Consider an object system and N unspecified particles. Observational 
determinations of the position of the object system must be relative to a 
physical frame of reference provided by the unspecified particles. Since 
they are unspecified, they have random positions in the hypersphere of 
radius R which constitutes space. The mean square deviation of a particle 
from the centre of the sphere is %R in each of the four coordinates; so that 
the mean square deviation of their centroid is ^R^N. Thus the physical 
frame of reference provided by the unspecified particles is such that its 
origin has a standard deviation %R/\fN compared with an ideally fixed 
geometrical frame. By the uncertainty principle the associated standard 
deviation of momentum is of order 2h^/N/R y or in mass units 2h\^N/Rc. 

Equivalently, the motion of each particle is a rotation about the centre 
of curvature of space and its wave function has a half-quantum h/2n of 
angular momentum in the plane of rotation, or a linear momentum hfirrR, 
the resultant of N such momenta in random directions is h^/Nf^trR. By 
either method we deduce a mass m^ of order hi/N/Rc as in (14-71), giving 
the energy of the physical reference frame referred to the geometrical 
reference frame. The observed relative motion of a particle referred to the 

l8-2 



276 Physical Applications [14-8 

physical frame is then apportioned between the particle and the frame 
according to the principles worked out in Chapters xi and xn, and the mass 
m p or m e attributed to the particle is given by the fundamental quadratic 
(14-51). 

It may be objected that by this procedure what is really the same com- 
parison mass ra is used over and over again for each particle in the universe. 
Quite so. That is the secret of the unexpectedly large masses attributed to 
the particles in quantum theory unexpected because the constant h which 
connects mass and curvature in quantum theory is at first sight much too large . 

In the exact determination of m in (14-71), several numerical factors, 
which might easily have been overlooked, have been introduced. We add 
some further remarks on these. 

The factor f arises as we have seen from the exclusion energy or gravi- 
tational potential energy of the particles, reckoned negative because the 
maximum exclusion energy or limit energy is our ordinary standard. For 
the present purposes N' excluding particles are equivalent to f N' non- 
excluding particles. The only point that arises is whether the same correc- 
tion should not have been used in (14-72). The answer is that (14-72) is a 
standard equation of relativity theory, here used to express our results in 
terms of the observed constant K. On referring to the derivation of this 
formula in general relativity theory, we find that M is the sum of the masses 
of the particles without deduction of gravitational potential energy (which 
is a non-tensor quantity). As we said at the beginning, our plan has been to 
solve the problem of a static distribution by two methods; (14-71) is the 
result of the solution by wave mechanics, and (14-72) the result of the 
solution by relativity theory. 

The factor f , introduced in passing from system A to system A', is perhaps 
less troublesome than the others, because it appears analytically and we 
are not directly concerned with its physical significance. We have seen, 
however, that it is due to the adaptation of the formulae to static problems 
in which the time dimension has specialised treatment, whereas the more 
elementary formulae are based on four coordinates to which Lorentz 
transformations are applicable. Alternatively we can regard it as due to the 
condition that, by considering a spherical space affixed radius JR, we have 
suppressed all waves in the radial direction. The actual R of the universe 
(referred either to an ideal geometrical frame or to a physical comparison 
standard) is, like all observables, subject to the uncertainty principle, and 
has a probability spread. But it is treated as fixed because, being the sole 
linear characteristic of the universe in its ground state, it is the standard by 
which all other lengths are measured directly or indirectly. This posterior 
fixity, conferred on R by its adoption as standard, conceals its intrinsic 
variability. 



14-8] The Cosmical Problem 277 

As soon as the static condition is relaxed, the variability of R (the local 
radius of curvature) forces itself on our attention; it is duly exhibited in the 
formulae of general relativity theory. In other dimensions, we pass from the 
non-static to the static condition by substituting constant standing waves 
for irregular wave motion. In the R dimension irregular wave motion is 
simply dropped, and is not replaced by standing waves. I do not suggest 
that this is an erroneous treatment; but we see how the fourth degree of 
freedom has come to be suppressed. 

To use four-dimensional waves in the present problem would defeat our 
purpose. The Einstein universe is unstable.! It is in statistical equilibrium 
only on the understanding that transfer of the energy of the three-dimen- 
sional waves into the radial dimension is inhibited. If transfer occurs, we 
have an expanding or contracting universe; presumably this is a state of 
pursuit of equipartition of energy an equipartition which can never be 
attained, since expansion of scale corresponds to a hyperbolic phase, and 
the wave functions in the radial dimension are exponentials instead of sines 
and cosines. 

Clearly it is more than a happy coincidence that the energy is multiplied 
by a factor f in passing from system A to system A' . So that before the 
division of the scalar particle into four elementary particles is actually made, 
a transformation equivalent to the substitution of four for three dimensions 
occurs. Perhaps it is better to express it the converse way; the four particles 
are effectively reduced to three before being packed into the static scalar 
wave function in three dimensions. 

Very difficult questions arise in regard to a factor 2 or 4; and it is only by 
the closest attention to the physical meaning of the analysis that we can 
hope to get this factor right. It is complicated at the outset because an 
erroneous factor 2 occurs in Dirac's theory through a confusion between 
double-valued and single-valued wave functions. It may be noticed that 
in the solution of the box problem (13-27) we take the momentum factor to 
be ife/27r whereas for the apparently similar standing waves here used we 
take it to be iA/7r. But in the box problem is an internal wave function 
describing relative motion as in the theory of the hydrogen atom; whereas 
here the scalar wave functions are referred to a geometrical frame. Further 
subtleties arise in the relation of scalar to neutral particles, in the repre- 
sentation of the particles of system B as an addition to, instead of a modifica- 
tion of, the particles of system A (so that when all the particles are specified 
we duplicate the universe), and in the liability to count the mutual energy of 
the particles twice over. The last correction was required in our second 
derivation of w , but not in the first derivation. I can only offer the solution 
here obtained as the best effort I can make to avoid these pitfalls with the 
t Perhaps the term "metastable" expresses the condition more precisely. 



278 Physical Applications [14-8 

feeling that it will be something of a miracle if I have really escaped them all. 
It is reassuring that the same value of N is obtained by an entirely inde- 
pendent investigation in Chapter xvi though even there a question of a 
factor 2 arises which is not easy to decide. 

Considerable light is thrown on these factors by a comparison with the 
theory of the Stern-Gerlach effect for which the corresponding factors 
can be checked observationally ( 12-8). In particular, the internal wave 
functions with four dynamical coordinates (comparable therefore with 
standing waves in four dimensions) are found to have a magnetic energy 
^ of that of a corresponding external wave function, whereas the single- 
phase wave function which defines m would (by the same treatment) 
have j 1 ^ of the energy. The 4 : 1 ratio is shown more directly in the 
magnetic energy than in the dynamical energy, because the masses assigned 
to the particles are such as to eliminate these factors and validate the 
current dynamical formulae which ignore them. The factor f shown in 
(12-86), which corresponds to the transition from strong to weak magnetic 
fields, is equivalent to that introduced in the present problem in the 
transition from disturbed to static conditions. 

I may add an explanation why it is necessary to treat the universe as 
composed of N f scalar particles rather than of 4JV' neutral particles in the 
main part of the investigation. A neutral particle has an equal probability 
of being positive or negative j^collection of 4JV' neutral particles has there- 
fore a probable charge eVW'. This is a large charge which would give 
a potential throughout the universe of the order 10 4 electrostatic units or 
10 6 volts. We cannot trust to probability to provide a neutral universe; we 
have to build it of units whose charge is definitely zero, i.e. scalar particles. 

It is worth noticing that in (14-74) the numerical factor J.-^f on the right 
has a simple relation to the numerical factor |.^(=|) on the left. The 
numbers 3, 10, 136 are the numbers of symmetrical components in 2-fold, 
4-fold and 16-fold matrices, respectively. I think it might be possible to 
exhibit the factor on the left as arising in the reduction of Dirac wave 
vectors to Pauli vectors (which are adequate for describing a static dis- 
tribution), in the same way that the factor on the right arises in the reduction 
from double wave vectors to simple wave vectors. 

14-9. The Recession of the Nebulae. 

Having found N, we can determine the Einstein radius B of the universe 
by (14-72). The result is 

jR= 1-234 . 10 27 cm. = 400-3 megaparsecs. 
The total mass and density in the Einstein state are 

M = 2-61. 10 55 gm. =* 1-32 . 10 22 x sun's mass, 
p e = 3-32. 10~ 27 gm. = 1 hydrogen atom per 500 cu. cm. 



14-9] The Cosmieal Problem 279 

The limiting speed of recession of distant objects is c/J?\/3 per unit 
distance. We obtain 

Speed of recession = 432 km. per sec. per megaparsec. 

The observational value of the speed of recession of the extra-galactic 
nebulae (usually given in round numbers as 500 km. per sec. per mp.) is in 
as close agreement as could be expected. 

In terms of the cosmical constant A the radius and mass of an Einstein 
universe are ~* 



In the mathematical theory of the expanding universe, it has been usual to 
treat three cases according as the actual mass M is greater than, equal to, 
or less than M e . The present theory leads to a different outlook; M is neces- 
sarily equal to M e . If we pay attention to relativity theory only, we have no 
ground for supposing that any static configuration of the matter of the 
universe can be found; therefore universes possessing no static configura- 
tions, i.e. with Jf> Jf e , have been considered possible. But when we treat 
the universe as a collection of N particles obeying the exclusion principle, 
we see that such a system necessarily possesses a ground state and therefore 
a static configuration; thus the universes without static configurations are 
inadmissible. 

I suppose that the ground state of the universe has hitherto been pictured 
as a highly concentrated "atom" with radius negligible compared to the 
present dimensions of the universe; so that it scarcely came to be considered 
in connection with the Friedman-Lemaitre theory. Our calculation shows 
that, on the contrary, it has a radius of 400 megaparsecs, and a density 
equivalent to 2 hydrogen atoms per litre. 

We thus discard the naive idea that in the beginning there was a cosmical 
constant; and that when the universe was made, the Creator had to decide 
whether the amount of matter created should be greater than, less than, or 
equal to the standard mass M e fixed by it. In the present theory A is of the 
nature of a constant of integration adjusted according to the actual mass of 
the universe. 

According to the Friedman-Lemaitre theory, if the universe started from 
the Einstein state of unstable (or metastable) equilibrium, it might either 
expand or contract. But since the Einstein state is the ground state, con- 
traction seems paradoxical. It would be interesting if a contracting universe 
could be ruled out in this way. But at present I cannot see that contraction 
definitely conflicts with wave mechanics; in comparing the ground state of 
the universe with the ground state of an atom we must bear in mind that 
the former is unstable and the latter stable. On the other hand, it seems 
impossible thai the contraction should continue indefinitely; and, if it 
cannot continue, it should somehow be prevented from starting. Formerly 



280 Physical Applications [14*9 

one supposed that the contraction might come to a natural end through 
quantum complications setting in when the particles became closely packed; 
but we now see that the " quantum complications" are already in full play 
in the Einstein universe. 

Although the relations between the natural constants have here been 
calculated for a special distribution of matter, they must hold for the 
irregular distribution in the actual universe. In determining the constant 
of gravitation experimentally, the physicist is not forbidden to arrange the 
matter in his laboratory in any way that suits the experiment; similarly the 
theoretical physicist is not forbidden to arrange the matter of the universe 
in any way that makes his calculation easier. In either case the value found 
for the constant will apply to all distributions, however widely they may 
differ from those used in the experiment or the calculation. Only we must 
take note that, in rearranging the matter to suit his purpose, the experi- 
menter cannot, and the theorist must not, violate any law of nature. Thus 
we have had to defend our rearrangement of the universe as a static con- 
figuration, by showing that the matter of the universe is necessarily such 
that it possesses a static configuration. 

These arguments would be complicated if we took note of the fact that, 
in addition to matter, the actual universe contains a certain amount of 
radiation. Since the amount is changing, it cannot be regarded as an essen- 
tial feature of the problem. It can, I think, be eliminated in the permissible 
"rearrangement" referred to in the last paragraph; but in any case the 
amount of radiant energy is trivial compared with the whole energy of the 
universe. 



CHAPTER XV 
ELECTRIC CHARGE 

15-1. Interaction . 

In classical physics an interaction between two particles means a difference 
in the behaviour of one due to the presence of the other. In wave mechanics 
we study probability distributions, and determine only probable behaviour. 
Interaction is therefore a difference in the probability of behaviour. The 
probability distribution of electron A , which specifies its chance of occupying 
a particular position or possessing a particular momentum, is modified by 
the presence of electron B. 

There are two ways of treating these changes of probability distribution. 
We represent the actual probability as the product of two factors: (1) the 
initial probability, or " basis of statistics", and (2) a modifying factor which 
incorporates any special information supplied. Analytically, the initial 
probability is the volume of the element of phase space containing the 
configurations considered; and the modifying factor is given by the product 
of the wave functions 0, <j>. The effect of the presence of electron B can be 
introduced either in (1) or (2). If we treat the presence of electron B as 
special information, its interaction is incorporated in the modifying factor. 
Alternatively we may regard it as a normal circumstance that the electron 
A, which we are considering, is one of several present in the region; we then 
incorporate the interaction in the initial probability and adopt a "new 
statistics'" for systems of two or more electrons. 

Both alternatives have been commonly employed in describing the inter- 
action of electrons. It is postulated that they repel one another with a 
Coulomb force. This tends to keep them apart, and thereby modifies the 
probability distribution which would have been attributed to them as 
independent systems whose probabilities combine by simple multiplication. 
It is also postulated that a system of several electrons obeys a new statistics, 
called Fermi-Dirac statistics. 

Both the Coulomb force and the Fermi-Dirac statistics describe an inter- 
action; that is to say, they assign to the electrons a probability distribution 
of position, momentum and spin different from the distribution for non- 
interacting particles, whose probabilities are independent and therefore 
combine by simple multiplication. But the Coulomb interaction is in- 
corporated in the modifying factor, and the Fermi-Dirac interaction is 
incorporated in the initial probability or basis of statistics. The Coulomb 
force changes the wave functions, so that they satisfy a modified wave 
equation containing an extra term called the Coulomb energy. The Fermi- 



282 Physical Applications [15-1 

Dirac interaction gives zero probability to the symmetrical wave functions; 
these are therefore omitted from the beginning, and the initial probability 
distribution of momentum is limited to such parts of phase space as corre- 
spond to antisymmetrical wave functions. Even if the two interactions were 
of independent origin it would be desirable to express them in more com- 
parable form. But I think it is obvious that current theory, by treating the 
interaction of electrons in this piecemeal way, has arbitrarily divided into 
two compartments a subject which is really one. It cannot seriously be 
maintained that the Coulomb force, which prevents two slow moving 
electrons from approaching one another, is an altogether distinct pheno- 
menon from the exclusion principle (contained in Fermi-Dirac statistics) 
which achieves the same result by forbidding 'them to occupy the same 
phase cell. 

It is not as though Fermi-Dirac statistics were intended to be a first 
approximation, giving the probability distribution in the limit when the 
electrons are so far apart that their Coulomb forces are negligible. The 
difference between classical and Fermi-Dirac statistics is only important 
when the electrons are crowded together; and the conditions in which we 
apply Fermi-Dirac statistics are precisely those in which the Coulomb forces 
are large. The attitude of current theory is altogether bewildering. It sets 
up an ideal scheme of statistics only to repudiate it (by introducing a large 
modification) in the very circumstances for which it is designed. 

This separation of the interaction of electrons into two effects strongly 
resembles the separation of gravitation and inertia in Newtonian mechanics. 
The latter taught that a body tends to move uniformly in a straight line 
by its inertia, but is pulled into a different path by the gravitational field. 
Similarly today quantum physics teaches that electrons tend to take up the 
probability distribution corresponding to Fermi-Dirac statistics, but are 
forced into a different distribution by their electrical repulsions. There is 
need for the same kind of unification of treatment that has proved so success- 
ful in the unification of gravitation and inertia. 

It is well known that Fermi-Dirac statistics arise from the indistinguish- 
ability of the particles concerned. If we are right in believing that Coulomb 
force is another aspect of the same interaction, it must also arise from the 
indistinguishability of the particles. To test this we must investigate the 
precise way in which indistinguishability modifies the enumeration of 
probabilities; so that we can determine its effect on the wave tensors, and 
hence on the wave equations which the tensors satisfy. The investigation 
is carried out in this chapter. We shall find that the effect of the indistin- 
guishability is to introduce an additional term in the wave equation, which 
turns out to be identical with that which has been adopted empirically to 
represent the Coulomb energy. 



15-2] Electric Charge 283 

The leading idea in the investigation is that the equations must be in- 
variant for interchange of the indistinguishable particles. Consequently we 
have a new kind of relativity transformation or "rotation" of the system, 
bringing about interchange. The interchange can be performed continuously 
a gradual transfer of probability from one identification to the opposite 
identification. The argument of the new transformation is called the per- 
mutation coordinate, and its conjugate momentum is called the interchange 
energy. On calculating the interchange energy, we find that it agrees with 
the observational value of the Coulomb energy. 

The foregoing argument was the actual starting point of the theory of 
protons and electrons developed in this book.f It was, I believe, the first 
introduction of the permutation coordinate and its conjugate interchange 
energy in wave mechanics. Now that interchange energy is regularly used 
in practical problems, it is difficult to see why the author's theory of the 
Coulomb energy of electric charges is still looked upon as a dubious excres- 
cence on wave mechanics. In the equations in current use the identity of 
interchange energy and Coulomb energy is accepted. J I do not understand 
why an investigation whose results have come to be admitted without 
question in the formulae in regular use is still commonly alluded to as a 
' * bold speculation ' ' . 

15*2. Interchange . 

The initial difficulty in calculating the effect of interchangeability is that 
interchange seems to be a discontinuous transformation. But wave mechanics 
has been successful in replacing quantum "jumps" by continuous analysis, 
and the same methods are available for treating the jump of interchange. 

Denote the coordinates, including suffix coordinates, of two particles 
(not necessarily indistinguishable particles) collectively by x, x'\ and let 
Y (x, x') be a wave function of the combined system. Let the operation of 
interchanging x and x' be denoted by Q, so that 

*V(x',x) = QW(x i x'). (15-21) 

If T is single-valued, $ 2 = 1. But the ordinary relativistic wave function 
has ambiguous sign ( 9-6). For wlien the corresponding space vectors *FX* 
are rotated through 360, so that all observable characteristics of the system 

t "The Charge of an Electron 1 ', Proc. Ray. Soc. A, 122, 358 (1929). 

j Dirao, Quantum Mechanics, 2nd ed., p. 228, equation (38). The interchange energy is 
given as JF ff {1 + (a r , a,)}, whose eigenvalue is the Coulomb energy V TB . The unitary matrix 
factor depends on the circumstances of the problem to be treated, and does not affect the 
identification. 

When T aj | (x^ 9 a/) is written as T (a? M , a; #/, j3), we call a, suffix coordinates. Suffix 
coordinates can take only the four values 1, 2, 3, 4. They are sometimes called spin co- 
ordinates; but this name should be reserved for angular coordinates conjugate to spin mo- 
menta. 



284 Physical Applications [15-2 

are unchanged, Y and X rotate through 180 and become Y, X. If the 
operation of interchange is treated as a rotation, repetition of the operation 
will bring us to the other branch of the double-valued wave function, so that 
Q2 T= _ Y Hence Q 2 =-l. 
We introduce an "extended wave function" denned by 

T(*,*',x) = e4xT (*,*') (15-22) 

Then, since # 2 =-l, 

'H*. *', x ) = (cos | x + <2 sin 



In particular T (x, x', n) = T (x' t x); and 

V(x,x' )X + *) = V(x',x, x ). (15-24) 

Thus the particles are interchanged by increasing x by IT. Intermediate 
values of x have a simple interpretation. By the usual rule (15-23) represents 
a superposition of the state Y (#, x') with probability cos 2 \x an( ^ ^he state 
Y (a/, a) with probability sin 2 #. The interchange is therefore represented, 
not as a sudden jump, but as a gradual transfer of probability from the 
original to the interchanged state, as x increases from to IT. This is the 
recognised method of treating transitions between discrete states in wave 
mechanics. 

Suppose that we are describing the distribution of a red particle and a 
blue particle. We use probability distributions to express our inexact 
knowledge of their positions. It is appropriate to provide in the same way 
for inexact knowledge of their colour. To take another example two golf 
balls have been driven at a short hole; there is a certain probability that 
there will be one ball on the green and one in the bunker; there is also an 
(unequal) probability that the one on the green will be your ball or mine. 
Therefore, besides stating the probability that there are particles at two 
points x, x', we can state the probability^ that the particle at x is the red 
one, or the probability 1 p that it is the blue one. This is provided for in 
the extended wave function Y (#,#', x) which includes a permutation 
coordinate x such that cos^x^JP- I* 1 general we treat a state represented, 
not by one value of #, but by a probability distribution over the coordinate 
X so that we have a probability distribution of what is itself interpreted 
as a probability. 

The wave function *F (x, x', x) primarily applies to the general case in 
which the two particles are distinguished with more or less uncertainty. In 
practice we confine attention to two limiting cases, namely definitely dis- 
tinguished, and entirely undistinguished particles. If the particles are 
definitely distinguished, # = or TT, and we can arrange that x shall always 
be 0. If the particles are indistinguishable observationally, x is an un- 
observable. In other words the rotation g = e* G x is a relativity transforma- 



15-2] Electric Charge 285 

tion to a different but equivalent frame of reference. This is an altogether 
new type of relativity transformation, which appears for the first time in 
double systems. But analytically it is of the same form as the well-known 
relativity rotations, and it has the same consequences. When x * s un ~ 
observable, the hamiltonian cannot contain x explicitly, but it will contain 
the momentum (if any) conjugate to #. That is to say, x is an ignorable 
coordinate. 

There is nothing mystical about the effects of indistinguishability. We 
do not suppose that an electron knows that it will not be distinguished from 
other electrons, and on that account conducts itself differently. We can 
imagine a being more gifted than ourselves who identifies each individual 
electron. He applies the ordinary equations of distinguishable particles to 
them, and his results are right; but his solutions do not interest us, because 
we can never obtain the observational data which he uses, and have no 
opportunity to apply or test his deductions. He observes, let us say, a red 
particle and a blue particle; he finds that the red particle is at x l at time ^ , 
and at # 2 at time t 2 , and deduces that it had a velocity (# 2 - 1 )/(^ 2 - $1) and 
a momentum w (# 2 -- a^)/^-^). We observe a particle at x l at time * lf and 
a particle at x 2 at time t 2 , but we do not know whether it is the same particle. 
Velocities of particles (and the corresponding kinematical momenta) are 
not observational data for us; and a system of dynamics which manipulates 
such data is useless for our purposes. 

The dynamical equations therefore depend, not on whether the particles 
are intrinsically distinguishable or indistinguishable, but on whether and 
to what extent they are in fact distinguished. 

We must distinguish the general wave function ^'(x.x'.x) which may 
be any function of its arguments from the particular wave function 
*( X >*'>X) defined in (15-22). The latter represents a uniform probability 
distribution in #. If the states corresponding to different values of x have 
different probabilities p x , their combined wave function is p^^(x 9 x' 9 x)l 
or more generally we can form a combination with different probabilities 
of different states T (x, x') for each permutation angle. 

Returning to the problem of the red and blue particles, the operators 
id fix, -id fix' do not give the momenta of the red and blue particles 
which are the momenta referred to in the dynamical equations for dis- 
tinguished particles. We can, if we like, regard -id/dx as the momentum of 
a composite particle, which has certain probabilities of being the red or blue 
particle. We have not the data required for applying ordinary dynamics, 
which does not profess to treat particles composite in this sense. But we 
can construct a formally similar dynamics by treating x as an additional 
coordinate; so that the whole momentum of the system includes, besides 
the momenta of two composite particles, a momentum of interchange. 



286 Physical Applications [15-2 

We have then to consider distributions of probability over the domain of 
x, x', x, and more especially to determine distributions which represent 
steady states. When account is taken of the spin components of the stream 
vector, the distribution is represented in phase space; and the ordinary 
136 -dimensional phase space of a double system is extended to 137 dimen- 
sions by the addition of a dimension representing change of g. We shall see 
later that the 137 -dimensional phase space is a development of the "aug- 
mented phase space" used in 12-3; so that from this point of view the 
permutation coordinate appears as compensation for the dropping of the 
relative time coordinate 2 ~*i when two systems are combined into one. If 
we have observational information e.g. an observation of colour which 
at any time distinguishes the particles with fairly strong probability, this 
will be represented as a wave packet in #; just as observational information 
as to their position is represented by a wave packet in x, x'. The wave packet 
will gradually disperse not because the particle is liable to change colour, 
but because we cannot trace definitely which of the particles whose colour 
we observed is the one which is now at x. 

From the general problem we pass to the two special cases. Firstly, when 
the particles are entirely undistinguished, the equations are simplified by 
the fact that x i 8 an ignorable coordinate and can be eliminated, leaving 
only a term in the hamiltonian which represents its conjugate momentum. 
Secondly, when the particles are completely distinguished, the equations 
are again simplified because x ^ constrained to be 0. This is more than a 
limiting case; it is a change in our point of view. We could not in an ordinary 
way prevent the wave packet concentrated at x = from dispersing. The 
constraint involves a change in the basis of statistics. An astronomer may 
inadvertently interchange the two components of a double star which he is 
observing; but he treats this as a mistake. He does not expect the laws of 
celestial mechanics to predict his " observational result". The constraint 
X = is imposed by stigmatising any other value of x as a blunder. We have 
seen that the dynamics of distinguishable particles cannot from its very 
nature be applied to indistinguishable particles; but why should not the 
dynamics of indistinguishable particles apply (so far as it goes) to dis- 
tinguishable particles? The answer is that theoretical physics is intended 
to agree with the experience of an observer who does not make mistakes, and 
will take varying forms according to the definition of what constitutes a 
mistake. 

Analytically, the probability distribution of distinguishable particles is 
limited to the 136 -dimensional section # = of the 137 -dimensional phase 
space. A change of the basis of statistics is made by limiting the initial 
probability distribution to 136 instead of 137 dimensions. Neglect of this 
distinction between the dynamics of distinguishable and indistinguishable 



15-3] Electric Charge 287 



particles has caused an error of a factor ff in many of the currently 
accepted formulae of quantum theory. 

From the present point of view protons are indistinguishable from electrons. 
It might be thought that the greater mass of the proton would be a dis- 
tinction impossible to overlook. But the mass is determined by the operator 
id/dt. We cannot differentiate the probability distribution of a particular 
particle until we have settled how to identify that particle at different times. 
Thus mass can never be used as a criterion for distinguishing particles; it 
presupposes that they have already been distinguished. 

The Principle of the Blank Sheet requires that at the start we should 
recognise no intrinsic distribution between the particles which we con- 
template, in order that we may trace to their very source the origin of those 
distinctions which we recognise in practical observation. The fundamental 
dynamics is the dynamics of indistinguishable particles; the dynamics of 
distinguishable particles is a practical adaptation to be used when we do 
not wish to analyse the phenomena so deeply. 

15*3. The Fermi-Dirac Law. 

In current theory it is usual to take the wave function Y of two particles to 
be single- valued. Then, if Q is the interchange operator, Q 2 = 1. In place of 
(15*22) we take an extended wave function 



T (x, x', x ) = 
Then e* i7 * = cos fr + iQ sin \TT = iQ 

and T (x, x', x + w) = Y (x' 9 x, x) as before. 
Consider the tensor transformation 



(15-32) 

where P is the interchange operator of the frames E^ , F^ of the two particles. 
Let their coordinates in three dimensions be x^ , x^', the time t being common 
to these and other particles. Their combined position is specified by a 
position vector 

) (15-33) 



in the double frame. We use the strain vector form of X, since the time is to 
be treated as invariant. A wave function of the two particles will be denoted 
indifferently by T ( Xfl , */) or Y (X). 

When the transformation (15-32) is applied, we have to consider, as in 
8-3, not only the direct change of T but the change of its argument X. 
Usually the transformation will introduce new matrices into X, so that 
*P (X) ceases to represent a distribution in the original 3-space. But when 



288 Physical Applications [15-3 

X = TT, we have g= P; and the transformation of the strain vector X is 
(since P is space-like) 



by (10-33). Thus we return to the original 3-space, but x^ and x^' are inter- 
changed. The transformed value of Y is 



by (10-37). Hence the combined result is 



*F' referring to the transformed distribution and *F to the original distribu- 
tion. Using the permutation coordinate, we can also write (15-34) as 

T (x, x', 0) = *F (x, x', IT) = e^- 1 ) T (a, *', 0) (15-35) 

by (15-31). 

Thus, in a sense, Q 1 = P- 1; so that we have found a possible form 
for the hitherto unidentified symbol Q. The actual relation is 

e iix?-D ^ |- e tfc<-p-i)] f (15-36) 

where the right-hand side is construed not as a simple multiplier but as a 
tensor transformation applied to the function which follows. (It is therefore 
not permissible to cancel out e~* f x on each side.) 

Let us now treat ^ as a dynamical coordinate, so that the transformation 
q gives an extended wave function representing a probability distribution 
over the seven-dimensional domain (x, x', #). The six-dimensional sections 
= 0, X = TT coincide with ordinary space (repeated for the two particles). 
If the wave function is single valued in ordinary space, Y (x, x', 0) and 
^(#,#',77) must agree; so that by (15-35), " = T. Hence, by (15-34), 



Thus the wave function is symmetrical for interchange of the particles. 
An antiaymmetrical wave function is obtained by the transformation 

g = ei f x(-^i) (15-37) 

which leads similarly to 



In this case (15-35) is replaced by Y' (x, x', 0) = -*F(o;,a;',7r); so that the 
transformation (15-37) is not a simple rotation in #. Belativistic rotation, 
or parallel displacement, in x generates a special wave function, which we 
have called the extended wave function; this is symmetrical. But, as we 
have pointed out (p. 285), the general wave function may be any function 
of the coordinates x, x' 9 #, subject to the usual conditions of single-valued- 
ness; and the antisymmetrical wave function is one of the general functions. 
Since it is not generated by parallel displacement in #, its covariant de- 



15-3] Ekctric Charge 289 

rivative with respect to x does not vanish. Denoting the antisymmetrical 
wave function generated by (15-37) by T, and distinguishing transforma- 
tion operators from multiplying operators by square brackets, we have 



(x, x' 9 x ) = &*<-*+] T (x, x') 

(a:,a 
, x') 



a; / , x ). (15-38)t 

The covariant derivative of Y a with respect to x * 8 contained in the factor 
[e**], since the covariant derivative of Y (x, x', x), which denotes as before 
the extended wave function formed by parallel displacement of Y (x t x') t 
is zero. 

The following remarks will perhaps make the significance of the 
mathematical procedure clearer. Consider a series of six-dimensional 
states V(x, x', a), distinguished from one another by a parameter a. 
In the special case in which a is unobservable this mode of division into 
states becomes degenerate, and the series can only be contemplated as 
a whole, i.e. as a seven-dimensional state. If a is promoted to the rank 
of coordinate, Y (x, x', a) is one such seven-dimensional state; but we can 
form other states p (a) X (x, x', a) by combining six-dimensional distri- 
butions with different (algebraic) probability coefficients P(OL). In the 
seven-dimensional state the probability fluid ( 81) is no longer restricted 
to flow in the planes a = const., and there will in general be a probability 
flux along the coordinate a, determined by the covariant derivative 
operator i8/Sa and therefore depending on jp(oc). This flux was not 
allowed for in the original equation of continuity of the probability fluid 
in six dimensions; thus X will not satisfy the same differential wave 
equation as X F. For given a, T (x, x', a) is a self-contained six-dimensional 
state, but X(#, x' 9 a) is not. In our application a is the permutation 
coordinate x ' an d the primary object of the investigation is to find the 
term in the wave equation for X, arising from flux in the x direction 
and depending on p(x)> which does not appear in the wave equation 
for MP 1 . To obtain its value we must determine p(x)- For an arbitrary 
angular coordinate a, the only limitatfan on p (a) would be that it must 
be periodic in a with period 2?r; but exceptionally p (x) has also to fulfil 
conditions at the half-period. We can interchange the particles, either by 
changing x to X + ^ *^ e wave function p (x) X (x, x', x) of the seven- 
dimensional state, or by interchanging x and x' in a six-dimensional 
section, and the two results must agree. Our analysis of this condition 

t The analytical part of the investigation is interrupted at this point. There is a direct 
continuation of it, starting from (15*38), in 15*7. 



290 Physical Applications [15-3 

can be put in the following form. We first ensure by our definition of 
X (x 9 x' 9 x) that its covariant derivative with respect to x vanishes ; that 
is to say, a change of the argument x represents only the effect of the 
relativity rotation provided by the indistinguishability of the particles. 
The test at the half-period then shows that, if X (x, x', 0) is a symmetrical 
function of x, x', no additional factor p(%) is required; but if it is anti- 
symmetrical, we must insert p (x) = [e*x]. Although p (x) is here expressed 
as an operator, we shall find later that it reduces to an eigenvalue e*x /187 ; 
so that the result is not inconsistent with our original limitation of p (a) 
to algebraic values. 

Our inability to distinguish observationally the separate six-dimensional 
states permits the probability to flow freely between them a flux which, 
if they could be distinguished, would be represented in a different manner, 
viz. by transitions. We do not here attempt to predict to what extent 
advantage will be taken of this freedom ; but we discover that the existence 
of a cyclic flux in the new direction will betray itself by its effect on the 
wave function for any of the six-dimensional sections. In particular, we 
have calculated the cyclic flux which is indicated when the flow in a six- 
dimensional section is that represented by an antisymmetrical wave 
function. 

It should be added that we have here adopted the solutions which give 
the smallest covariant derivative. There exist also solutions p (x) = [e 2ni *] 
for symmetric wave functions, and p(x) [d 2n+ *** x ] for antisymmetric 
wave functions. The ultimate effect of taking w = is that we shall deter- 
mine the minimum charge of a particle. 

According to the principle of Fermi and Dirac, the wave function of two 
elementary particles is antisymmetrical. Their theoretical treatment only 
went so far as to show that it must be either symmetrical or antisymmetrical; 
the choice of an antisymmetrical function depends on empirical considera- 
tions. We cannot at the moment go further than they did. Accepting their 
conclusion, the wave function for a pair of elementary particles must be 
taken to be T. 

We must now consider the "relativity of identity ". We have stressed the 
fact that a simple wave function $ (x) yields nothing observable, since it 
contains no reference to any comparison object for measuring x. This 
objection is removed in the double wave function T (a?, #'), since one particle 
can serve as comparison object for the other. But the objection reappears 
in Y (x, x', x)> since there is no comparison object for x- Just as we require 
two particles to provide observable differences x^x^, we require four 
particles a quadruple wave function to provide observable differences 
X #'. An absolute permutation coordinate referred to an abstract frame 
has no observational meaning, even when the particles are distinguishable. 



15-4] Electric Charge 291 

We may observe certain distinctions, but the interpretation which we place 
on these distinctions is not a matter of observation. We have temporarily 
shelved this difficulty by identifying the particles by a characteristic 
(colour) recognised in some supernatural manner; but actually colour might 
be changed by Doppler effects, and it is no more an absolute criterion of 
identity than position.f All we can do is to lay down a scheme of equations 
to be conventionally adopted as a criterion that x * s constant, and then 
measure changes of x relative to this standard. 

Standing waves and progressive waves aflf ord an example of the relativity 
of identity. We have seen that the particle represented by a standing wave 
is of composite identity (13*1). That depends on the usual view that a 
progressive wave represents a particle with single identity throughout all 
time. But it is equally possible to define the particles represented by 
standing waves as the real individual particles; then the progressive waves 
represent composite particles. 

Accordingly a system of two particles must be referred to a comparison 
fluid described by double wave functions and involving a comparable 
permutation coordinate XQ The angles x &nd XQ being referred to an abstract 
"frame of identity" are unobservable; but the probability distribution of 
the relative angles x XQ > contained in the quadruple wave function of the 
double object system and double comparison fluid, may have observable 
characteristics. The partition of the relative permutation angle into 
" absolute " permutation angles referred to a frame is an analogous problem 
to that which led to the theory of the Biemann-Christoffel tensor. 

From this point our investigation bifurcates into two problems. Firstly 
(15- 4-1 5- 6), we calculate the interaction of two particles forming a hy- 
drogen atom. Secondly ( 16-7), we calculate the interaction between pairs 
of particles in any assemblage. J There is a considerable difference between 
the methods used; and the fact that both lead to the same value of the 
Coulomb energy is a useful check on the details of the calculation. 

15-4. Interaction in the Hydrogen Atom, 

We shall now determine the interaction between two undistinguished 
elementary particles which form a steady system. We know that, for a steady 
state to be possible, the particles must be of opposite sign. The system is 
therefore identified with a hydrogen atom. 

f Consider a double star with equal components, having a very large orbital velocity in 
the line of sight. We can distinguish the two components by observing that one is red 
and the other blue. Half a period later we can again distinguish the red star and the 
blue star. But the stars thus identified do not obey the accepted laws of celestial 
mechanics. The accepted laws are obtained by identifying the blue star at time t with 
the red star half a period earlier. 

J Atomic nuclei are not considered. 

19-2 



292 Physical Applications [15-4 

As in 12-6, we re-resolve the hydrogen atom into an external and an 
internal particle of masses 

.i/-^ fl -m , /t=ii 6 m . (15-41) 

The time direction is as usual taken to agree with the momentum vector of 
the external particle. Interchange of the proton and electron does not 
affect the external particle. We therefore confine attention to the internal 
particle of mass p. Its coordinates f l9 2 , 3 are the relative coordinates of 
the proton and electron, and are reversed in sign by interchanging them. 
The change is most simply described by using angular coordinates, so that 
the displacements and momenta are rdO^, id/rdO^. Radial displacement 
and momentum are treated in the same way by Letting 6 f = log r . The inter- 
change is then equivalent to reversing the sign of r, leaving the angular 
variables unchanged. 

It may be well to repeat that the interchange is purely subjective as 
when an astronomer inadvertently interchanges the two components of a 
double star, and so publishes a position angle wrong by 180. By such a 
mistake relative coordinates are given the wrong sign; only in treating 
indistinguishable particles we do not count it as a mistake, since there is no 
criterion for deciding which is the right sign. 

Continuous interchange is represented by taking r = r e**; or if necessary 
we may take r = r e tt x, where Q is any symbolic square root of 1. 

The states of the internal particle will consist of probability distributions 
over the space coordinates f l9 f 2 , 3 and the permutation coordinate x- 
There is no time coordinate in an internal state. 

It would not be illegitimate to treat states in which the distribution is over 
i> 2* fa> on ty> with x a constant for the state. It is purely a question of 
practical application. As explained in 15-2 the resulting system of 
dynamics would be true, but useless to an observer who (owing to inability 
to distinguish the particles) could not obtain the data required for its 
application. Every angular coordinate has to be treated as we here treat ^; 
if the conditions are such that it becomes unobservable, a degeneracy is 
introduced, and the states which would have been distinguished by different 
values of 6 are run together into a single state. The ultimate reason for this 
is that in physical applications we have to take account of transitions (due 
to external perturbations) between the states given by different values of 0; 
as 6 approaches degeneracy these transitions become more frequent and the 
calculation becomes unmanageable; for completely degenerate 6 an in- 
finitesimal perturbation is sufficient to cause transitions, and the states can 
only be treated as a combined whole. Conversely, the transition causes only 
an infinitesimal perturbation of external systems and is therefore un- 
observable; if we cannot detect a transition, we cannot distinguish the states. 



15-5] Electric Charge 293 

We have to picture, or represent symbolically, a rotation which changes 
r continuously into r. For a picture we require an extra-spatial dimension 
a through which r can turn. For symbolic treatment a matrix or symbol E a 
must be associated with a in the same way that E l9 E 2 , E 3 are associated with 
f i > f 2 > f s Owing to the absence of a time coordinate, a symbol J 4 is standing 
idle; this is available to represent a dimension perpendicular to E^E^E^, 
and we therefore set E a =E^. The question whether E is really the missing 
EI does not arise; it is sufficient that its commutation relations with all 
other symbols in the formulae are identical with those of JS? 4 .f If E r is the 
matrix associated with the direction of r when x = 0, a point in the four 
dimensions will be represented vectorially by E r r + E^a, and the x rotation 



The spread of the probability distribution over the additional coordinate 
a introduces a corresponding term E^ d/da in the differential wave equation. 
Or, using the standard form (8-631) for a strain vector, we have 

" - (15>42) 

The extra permutation dimension thus plays the same part in an internal 
state as the time dimension plays in an external state. 

Except that we see that the energy of an internal state arises from inter- 
change, being conjugate to the linear interchange coordinate or, our only 
result thus far is to obtain the form of wave equation, already familiar for 
external particles. The equation expresses the conservation of probability; 
but the coordinates, etc., contained in it are unobservable, being referred 
to the frame and not to the comparison fluid. We have next to derive from 
it an equation suitable for practical use. 

15-5. The Fine Structure Constant. 

It is necessary to consider the internal particle in conjunction with its com- 
parison fluid. Since m = 136/x, the observable angular displacements 0' are 
analysed into a displacement = \\\0' of the particle and a recoil = ^9' 
of the partial comparison fluid. The question now arises whether the same 
partition applies to displacements of the permutation coordinate #. We see 
immediately that the partition of x is governed by quite different consider- 
ations from the partition of 6 which is based on the theory of the Biemann- 
Christoffel tensor. 

In the first place there is no need to admit any recoil of x- The comparison 
fluid is an idealised substitute for actual reference objects; and we are free to 
choose either a "distinguishable" or an "indistinguishable" comparison 

f The use of rectangular relative coordinates implies that the region considered is small 
enough to be treated as flat, so that E 6 also is idle. But this absence of E 6 is a casual feature 
of a special problem; it is not comparable with the enforced absence of E it arising from the 
definition of an internal state as a simultaneous state. 



294 Physical Applications [16-5 

fluid, i.e. fluids whose constituent particles are respectively distinguishable 
or indistinguishable. Inasmuch as the fluid replaces distinguishable macro- 
scopic bodies, it is a less violent distortion of the natural conditions to use a 
distinguishable comparison fluid. Then #0 is constrained to be zero, and 
there is no recoil. 

There is no breach with our previous conventions and definitions in 
admitting this exception, though it will complicate subsequent applications. 
Initially we were free to specify the recoil in each coordinate arbitrarily; 
but later we introduced the condition that the (negative) recoil in the time 
direction is such that the object particle and the comparison fluid move 
forward together in time. Relativity conditions then determine the recoil 
in any other direction which can be connected with the time direction by a 
relativity rotation. But the direction x & defined by a tensor of different 
rank, and is not connected with the other directions by a relativity rotation, f 
It would be more of a breach with our previous conventions to admit an 
indistinguishable comparison fluid; for in all previous references to the 
comparison fluid we have treated it as composed of distinguishable particles. 
In particular its connection with the metrical tensor g^ v has been fixed on 
that basis. To substitute an indistinguishable comparison fluid at this stage 
would derange the metric, and upset our standard equations. 

Nevertheless it is instructive to consider what would be the result of 
employing an indistinguishable comparison fluid. In actual phenomena, 
shown by a comparison of the microscopic object particles with macroscopic 
reference objects, the comparison fluid is eliminated, and the final results are 
unaffected. The use of an indistinguishable comparison fluid as intermediary 
will simplify the microscopic part of the theory at the expense of complicat- 
ing the macroscopic part of the theory of these phenomena. The macro- 
scopic theory is affected because the energy tensor of the comparison fluid 
determines the metric to which the actual measures of length are supposed 
to be referred. We have seen that continuous interchange is represented by 
the transformation r->rex. This transformation may be attributed to an 
absolute change of r, or to an absolute change of the standard of length. 
(Since the other variables are angular, r is the only quantity affected by a 
change of the standard of length.) The standard of length for the object 
system is contained in the comparison fluid the idealised substitute for 
metre rods, etc. Hence the partition of x' into x an d Xo corresponds to a 
partition of the apparent change of r into absolute changes of r and of the 
standard gauge. That is to say, the "recoil" of the change of r is a gauge 
transformation. 

f We now propose to represent it as relativistically connected in a four-dimensional 
picture, assigning to it the matrix E^E r ; but the condition of admissibility of this repre- 
sentation is the point we are now considering. 



15-5] Electric Charge 295 



For a recoil Xoiirx'f the gauge is modified by the factor e *' 187 and 
becomes complex (if Q = i) or matricised. If we are to pursue this method we 
must abandon Riemannian space and adopt WeyPs geometry which admits 
complex gauge transformations, or the author's extension of it which admits 
more general transformations of gauge equivalent to matrix transformations. 
In wave mechanics we prefer to keep to Riemannian space; then x can have 
no recoil; and the equations of the microscopic theory are complicated by 
the fact that x behaves differently from the other angular coordinates in 
this respect. Before studying the phenomena which result, we notice two 
points of interest: 

(a) The equivalent complex gauge transformation corresponds to an 
electromagnetic field (8*8). Thus the phenomena will be of electro- 
magnetic character. 

(6) The gauge would have been changed by a factor eWis? ( or by a ma trix 
having this eigenvalue); thus a coefficient 137 is introduced. This coefficient 
is known empirically as the fine-structure constant. 

We have seen that formally a takes the place of an imaginary relative 
time f 4 . We shall try to elucidate this connection. Let (x l9 x 2 , x z , t), 
($1, x 2 ', # 3 ', *') be the coordinates of the proton and electron, and let 
(f i > f 2 9 3 > T ) be the differences x x l , etc. By the definition of a combined 
system r is constrained to be zero. We may, however, regard a displacement 
dr as a transformation to a frame of reference with a different reckoning of 
simultaneity. Consider the transformation 



The interval from the origin to the point considered is (r 2 r 2 )*=r ; and we 
can show easily that the intervals between all other pairs of points are like- 
wise independent of u. Thus a physical system occupying the domain 
(i> &> fa) * s intrinsically unaltered by the transformation. Apparently all 
distances in it are expanded in the ratio coshw; but this is accounted for by 
the fact that a new reckoning of simultaneity has been introduced, which 
antedates each particle by a time r = r tanhw, proportional to its distance 
from the origin. 

The corresponding transformation for imaginary relative time q is 



r = r ex, a = r 
For small displacements from the zero state (u = 0, x = 0) we have 



Thus far our formulae refer to changes of the system of reference real 
or imaginary changes of reckoning of simultaneity. But we can employ the 
transformation in the usual way to define a series of states of a system in the 
same frame of reference. The states will represent the same system ( 19 2 , ,) 



296 Physical Applications [15-5 

expanded in the ratios cosh u and e*x respectively, the latter expansion being 
interpreted as in 15-1, as a change of relative probability of the direct and 
interchanged states. The u series of states is unclosed (hyperbolic trans- 
formation) and a distribution of probability over it could not be represented 
in phase space. That is to say, in analysing the total distribution of pro- 
bability over the domain (f l , f 2 , 3 ) , into the sum of a number of elementary 
distributions, we cannot accept a dissection into a series of states con- 
tinuously varying in scale. The corresponding expansion of the wave func- 
tion in a series of elementary wave functions would not be convergent. For 
that reason, when once we have fixed our frame of reference, the u trans- 
formation is excluded, and r is identically zero. But in saying that an internal 
state is formed by the simultaneous configuration of two particles, we refer 
to simultaneity in real time; they may be non-simultaneous in imaginary 
time if we can find a meaning for such a phrase. Thus the % series of states 
is admissible (as a mode of dissection of the total probability distribution); 
and since it is a closed series (circular transformation) a distribution of 
probability over it can be represented in phase space. 

This clears up a point which personally I have found most difficult. It 
had seemed to me that for a real displacement dr, just as for a displacement 
dx> there would be no recoil of the comparison fluid.f An increase of r to 
r + dr might be attributed to an absolute change of r or to an absolute change of 
the standard of length, the latter constituting the "recoil" of the change of r. 
But since Riemannian geometry excludes changes of gauge, we can admit 
no such recoil. The argument is the same as that which led us to exclude 
recoil of x which would represent imaginary change of gauge. In short, 
Riemannian geometry requires a comparison fluid rotatable in every direc- 
tion, but not expansible or subject to interchange of its particles. 

To see the fallacy of this argument, we must recall that the same dis- 
placement dgp may be produced by a variety of rotations in different planes, 
corresponding to different transformations of the wave vector. Therefore 
when we treat a displacement dr or a momentum i djdr, we must not jump 
to the conclusion that it corresponds to a u transformation. By reference 
to the mode of formation of the wave equation, we see that the r-momen- 
tum which appears in it is the E r5 component of a space vector, and corre- 
sponds to rotation about the centre of curvature of space-time; this, of 
course, produces the usual recoil of the comparison fluid as provided for in 
the R.C. tensor. 

The nature of a displacement d^ at any point whether it is a rotation 
about the centre of space-time, or about an origin in the domain (^ , 2 , 3 ) , 
or an expansion of scale cannot be immediately seen by inspection. It 

t If the argument were correct, the Coulomb energy term found in 15*6 would be 
duplicated. 



15-6] Electric Charge 297 

depends on the system of analysis into elementary states furnished by the 
four commuting operators W, U l9 U 2 , Z7 3 . But without entering into the 
details of this analysis, we can show that there will be only one coordinate x 
which produces no recoil. The argument above was that whatever applies 
to the imaginary transformation of r should also apply to the real trans- 
formation of r ; but this is forestalled by the earlier result that of two anti- 
thetic transformations one will be circular and the other hyperbolic, and 
only the circular transformation is admitted in our system of analysis into 
elementary states. Thus we can only be concerned with one of the two 
transformations. 

We may recall that an E^ transformation displaces positive and negative 
charges in opposite directions in neutral space-time, and is therefore asso- 
ciated with polarisation rather than simple translation (6*3). It is this 
aspect of the r displacement as a change of separation of positive and 
negative charges which is handled in the internal wave equation. 

15*6. The Coulomb Energy. 

In the wave equation (15-42) the internal particle is represented by a simple 
wave vector $. The justification for separating \ft from the double wave vector 
of the particle and comparison fluid, and treating it as an independent 
distribution, depends on the theory of Chapter xn; but no account was then 
taken of the permutation coordinate. To rectify this we must go back to the 
quadruple wave function of the proton and electron and the corresponding 
unspecified particles which form their comparison fluid. This is resolved 
into two interchangeable double wave functions; or into a double wave 
function of an external particle (without permutation coordinate) and 
comparison fluid, and a double wave function of an internal particle (with 
permutation coordinate) and comparison fluid. Considering the latter, the 
permutation coordinate raises the number of dimensions of the phase space 
to 137. If it behaves symmetrically with the other dimensions (which implies 
that there is the same recoil in x as in the other coordinates) the same theory 
of resolution into simple wave vectors applies, except that the ratio of the 
mass [L of the internal particle to the mass w of the comparison fluid is 
now given by 13 7/ = m (15-61) 

instead of 136/x = m . 

Hence, if the comparison fluid recoils in #, the wave vector r satisfies the 
wave equation of the form (15*42) 

' =o - (15>62) 

But our standard comparison fluid does not recoil in x, and we wish to find 
the wave equation of the corresponding wave vector 0. 



298 Physical Applications [15-6 



In ( 1 5- 62) a displacement da = irdx is accompanied by a recoil d^o 
of the comparison fluid, in accordance with the relation w ^Xo = //^X- To 
eliminate this recoil we must transform the frame of reference so that the 
permutation coordinate of the comparison fluid in the new frame remains 
constant. The required rotation of the frame is in the forward direction of 
Xo, since the "recoil" is really an advance (owing to the time-like character 
of the matrix associated with it). The transformation due to the rotation of 
the frame is therefore 



for small values of a. (15-63) 

The transformation matrix is properly the matrix F r4c associated with the Xo 
rotation of the comparison fluid in its own frame F^\ but, since _F r4 commutes 
with all the symbols in (15-62), it is for our purposes an algebraic square 
root of 1. Substituting in (15-62), we have 



so that (putting a=0 after the differentiation) 

-' (16 ' 64) 



which is the required wave equation. 
Comparing with (9-64), we have 



Hence the value of the fine structure constant is determined as 

(15-65) 



15*7. Interaction in Systems of Particles. 

We turn now to the general problem of interaction in an assemblage of 
elementary particles. The total interchange energy of a particle will arise 
from interchange with every other particle. The difference from the previous 
investigation is that, since the particle has to be paired with more than one 
other particle, the special coordinates x^ , ^ are inappropriate; and we have 
to obtain expressions in terms of the coordinates x^ , x^ referred to an arbi- 
trary origin. 

Considering a wave function T of two particles, we resume the investiga- 
tion in 15-3 at equation (15-38), where it was broken off. In that equation 
the operator etow-D O r [e^x(-^-] gives the change of T due to parallel 
displacement in x- (Displacement in x is, for indistinguishable particles, a 
relativistic rotation of the frame of identification, and these operators give 



15-7] Electric Charge 299 

the nominal change of T due to its being referred to the rotated frame.) For 
antisymmetrical wave functions there is by (15-38) also a "real change" of 
Y represented by the operator [e*x]. The co variant derivative 8/8^ measures 
the real change; that is to say, 8/8# for Y is equal to dfix for [e*x]. But [ft*] is 
not a simple multiplier; it turns Y (X) into e'x Y (e 2i *X). We shall show that 
the change of argument eliminates ^|f ths of the factor e*x. 

When X-+e 2i *X, the coordinates become e 1 **^, e 2 '*^'. Since Y repre- 
sents a probability distribution over real coordinates, its interpretation for 
a complex argument requires definition. We are interested only in the 
co variant derivative of [e**]; therefore any part of it which represents 
parallel displacement of the function Y (z, x' 9 x) on which it operates may 
be ignored. A parallel displacement is produced by applying a tensor trans- 
formation q = e i( * to all the symbols concerned; this yields 



where x' is the transformed value of #, whose tensor character we leave for 
the moment undetermined. We therefore define Y for a complex argument 
oftheforme 2 *X:by 

ei*^(&i* x ,e,K*x',x') = ^(x>x'>X)> (15-71) 

since the two expressions are in any case equivalent for the purpose required. 
We want, however, to find Y(e 2 * a #,e 2l ' a a; / , ^); for a will ultimately be put 
equal to x> an d we naturally do not apply the transformation to the argu- 
ment which defines the transformation.! 

The transformation of the double strain vector of the distribution can be 
treated as a uniform gauge transformation JR-> Retf of the radius of the 
137-dimensional phase space. Denoting the strain vector YO* by S, the 
first effect of the transformation is to multiply S by e 137 *0 on account of the 
change of measure of the volume element attached to it; and the second 
effect is to multiply it by e~ 18W on account of the renormalisation necessary 
through the whole volume of phase space becoming e 187 #Q. These are 
equivalent to the two compensating changes in (15-71). It does not much 
matter which way round we identify them; but presumably the first is the 
result of the change of coordinates in the argument, and the second is the 
direct tensor transformation of Y and O by a factor e* a , which gives a factor 
e 2ia in 8. Thus 2a= - 1370. 

If we exempt the coordinate x fr m ^e transformation, so that the 
volume element contains dx instead of e*Pdx, *^ e ^ ra ^ factor becomes e 138 #. 
The exemption is a simple matter, because the interchange matrix P com- 
mutes with all the space-like matrices in the (?Z>-frame, i.e. the frame in 

t That would lead to the same confusion as the statement "every number on this page 
should be divided by 2", which (being amended in accordance with its own instruction) 
implies that the numbers should be divided by 1, and therefore that they should be divided 
by 2, etc. 



300 Physical Applications [15-7 

which the symmetrical and antisymmetrical wave functions are separated 
( 10-6). The second factor e~ 137l ' is unaffected. The complete cancelling only 
occurs when the transformation (parallel displacement) is applied to all the 
quantities concerned; in withdrawing one of them from the transformation, 
we do not have to introduce any compensating change elsewhere. The 
resultant change of S is therefore e i36tf. e -i37tf ^ta/ia? The change of T or 
O is e a/i3 7 . Accordingly, putting a = x, 

[e'*] = e*x/ 137 . (15-72) 

Hence the angular momentum is 

-iS/8x= 1/137. (15-73) 

To summarise the argument from the beginning: Indistinguishability of 
two particles causes a degeneracy, which is treated (like other cases of 
degeneracy) by introducing a dynamical coordinate #. The angular momen- 
tum i 8 fix is quantised by the condition that Y shall be single-valued. The 
change of T with x may be simple parallel displacement, in which case there 
is no angular momentum;! or there may be an additional real change. But 
unlike other cases of quantisation, the condition of single-valuedness is 
applied after a half-period when the original distribution has reappeared 
with the particles interchanged; and the Fermi-Dirac condition determines 
the real change to be that expressed by the transformation operator [e**]. 
If a transformation [e /a ] were applied to *(X, x), X being transformed in 
accordance with its tensor character as well as T and X, the result would be 
a parallel displacement which contributes nothing to the angular momen- 
tum. In the present case the transformation [e l 'x] is not applied to the co- 
ordinate x whose value it fixes. Consequently, instead of the factor efx being 
completely neutralised by the change of argument of Y, 1 out of the 137 
coordinates of phase space is left untransformed, and fails to do its share 
in neutralising e l X Thus a factor e** /137 survives. 

We have now to insert in the wave equation an extra term representing 
the linear momentum which corresponds to (15-73). We recall that the 
linear coordinate a corresponding to the x rotation is the imaginary relative 
time of the two particles. So that, if r is the distance between the two par- 
ticles, dardx- Let iK r be the matrix (in the double frame) associated with 
the direction of r.% It will be a y matrix, which by (10-35) commutes with P, 
and is therefore unaltered by interchanging the particles and measuring r 
in the reverse direction. The matrix of the x rotation, given by (15-32), is 
i(P+l). The matrix of the direction of a is the product K r (P+l). 

t Other than the quantum, which is added in our usual non-relativistic treatment, and 
dropped again when the angular momentum is converted into linear momentum. 

J We take iK, so that the direction may be associated with a symbol whose square is 1, 
as in a simple frame. It is understood that the strain vector representation is used throughout. 



15-7] Electric Charge 301 

The new momentum to be inserted in the wave equation is therefore 
(of. (8-37)) 



This is divided into two parts. The term K r j\yir embodies the reduction 
from the metric naturally associated with indistinguishable particles, whose 
initial probability distribution extends over 137 dimensions, to the standard 
metric associated with distinguishable reference objects. If there are two 
particles only, one of which is taken as the origin of polar coordinates, so 
that the hamiltonian contains a term (iK r )( id/dr), we can amalgamate 
jfir r (3/ar+l/137r)into# r 8/8r by setting Y = r~ 1 / 137 ''Y / , and taking Y' as the 
new wave function (cf. the treatment of (8-38)). But since we have already 
treated the case of two particles more directly, it is unnecessary to pursue 
this further. It is sufficient to notice that the term K r /I31r is the equivalent 
in the general formula of the change of mass from /A to /x' in the internal wave 
equation of the hydrogen atom, which will be considered further in the 
next section. The remaining term represents the Coulomb energy: 

K P 

Coulomb energy = -- . (15-75) 



The main additional result is that the matrix associated with the Coulomb 
energy is K r P. We naturally associate the Coulomb energy (or more strictly 
the Coulomb momentum) with the direction of r; we now see that it is 
necessary to multiply the matrix giving the direction of r by the symbol P 
in order to obtain the direction of the Coulomb momentum. We may there- 
fore describe the Coulomb momentum as equal to P/137r in the direction 
of r. In current perturbation theory it is taken to be P x /137r, where P l is 
given by (10-385), the other factor of P being omitted. How far this omission 
is justified will depend on the nature of the application. 

Light is thrown on the occurrence of P by the transformation investi- 
gated in 10' 8. On multiplying matrices in the UP-frame by P, we obtain 
matrices in the G^-frame. Thus the factor P disappears when we adopt the 
GtfMrame, i.e. when we divide the system of two particles into external and 
internal wave functions. Conversely we might have anticipated that the 
purely algebraic Coulomb energy, found in the internal wave equation of 
the hydrogen atom, would acquire a matrix P when referred to the frame 
of the coordinates # , x'. 

The previous investigation treated the interaction between particles of 
unlike sign; the present investigation applies most obviously to particles of 
like sign. We must suppose that E^ and F^ are both right-handed or both 
left-handed frames; otherwise there is no interchange operator P. But since 
no assumption has been made as to the nature of the stream vectors of the 
two particles, they may well have charges of opposite sign. It would, I think, 



302 Physical Applications [15-7 

require deeper analysis to show that the sign of the energy depends on whether 
the charges are like or unlike. In this section we have been content to use the 
Fermi-Dirac principle (partly theoretical and partly empirical) to obtain 
the operator \ffx\\ the principle is equally consistent with [>-**], so that the 
sign of the energy is left undetermined. It is possible, however, to go behind 
the Fermi-Dirac principle and, by investigating the quadruple wave func- 
tion, trace the Coulomb energy to the absence of recoil of the x coordinate 
in the comparison fluid, as we have done in the special case of a hydrogen 
atom. I think it would not be difficult to verify in this way the dependence 
of the energy on the sign of the charges. 

15-8. The Factor -JfJ. 

A new point brought out in this investigation is that the mass concerned in 
the internal wave equation is not p, but ft', where, by (15-61), 



In considering the consequences of this, we must remember that //, does not 
represent an energy actually present. The mass of a hydrogen atom on the 
verge of ionisation is M ; and although, for the purposes of analysis, an 
internal mass p. or p! is added, it is subtracted again at the end of the in- 
vestigation. When the atom is in a lower quantum state, ft is involved as a 
coefficient in the energy tensor ft-^/S^S^ of the internal wave function. 
When ft is reduced to ft' the energy tensor is increased, and the energy 
differences between the different states are increased. The actual energy 
differences are therefore ^f| times greater than those calculated in the usual 
way from Sommerfeld's formula (9-372), which ignores this factor. f 

It will be seen that the decrease of ft has just the opposite effect to that 
which we should at first have expected. The explanation is that ft is not under 
any circumstances a rest mass, but occurs only as a divisor in the expression 
(w^ + w 2 2 + ro 3 2 )//i for the kinetic energy. In particular it is not permissible 
merely to substitute ft' for ft in Sommerfeld's formula. 

The following alternative treatment leads to the same result. Consider a 
proton and electron confined within a rectangular boundary, so that there 
is no angular momentum, and the steady state can be analysed into standing 
waves in three perpendicular directions. We take the energy to be either 
positive or so slightly negative that the size of the atom is comparable with 
the dimensions of the enclosure. Equating the energies of the proton and 
electron to the energies of the external and internal particles, we have, by 



The Coulomb energy is omitted. 

t In (9-372) the symbol /* is used for a quantum number, and the internal mass 
v +mt) is denoted by m. 



15-8] Electric Charge 303 

Subtract from each term the energy m of a comparison particle, so as to 
leave only the energy due to specification. Then 



(15-822) 
where W ' = w +/i=ffw (15-83) 



by (15-41). 

Thus in order that the external and internal energies may be treated as 
additive in the same way as the energies of the proton and electron, the 
internal energy must be referred to a comparison mass m f which is |f times 
the comparison mass for External wave functions. The current treatment 
replaces (/*-f ci 2 /^- w ') by a linear hamiltonian, taking it for granted that 
the theory developed for external wave functions applies. We have seen 
that it is necessary to decrease /z to ^', which is equivalent to increasing the 
comparison standard from ra to ra ', so far as the internal state is concerned. 
But when we introduce the condition that the energy of specification of the 
atom shall be the sum of the energies of specification of its internal and 
external states, (15*822) shows that the right course is, not to diminish ju, 
to p in the current (Sommerfeld) formula, but to treat the formula as 
expressing the energy in a unit which is W '/w times the unit used for the 
external energy. 

Ideally we can measure experimentally the energy e /x of a particular 
quantum state of the hydrogen atom, and hence determine the mass /u by 
Sommerfeld's formula. The external mass M can be found experimentally 
by some procedure equivalent to counting the number of atoms in a quan- 
tity great enough to be weighed macroscopically. From M and /x, m p and 
m e can be calculated. But the ordinary determination of the masses by this 
method, which omits the factor ^ff , will be incorrect. The observed energies 
-/*, or the observed energy differences between different quantum levels, 
must be decreased in the ratio f|f before being compared with Sommer- 
feld's formula. 

Since M is very nearly the mass of the proton and p, is very nearly the mass 
of an electron, the practical effect of neglecting the factor is that the so- 
called observed mass of the electron is ff | times too great, that of the 
proton being correct. It was first pointed out by W. N. Bondf that the 
observed values of the various constants would come into line with the 
author's theory, if it could be assumed that the observational determinations 
of e/m e really determined Hf e / m e- At that time I was aware that a 
factor ||f would be involved, but had not been able to determine its precise 
incidence. 

t Nature, 133, 327 (1934). 



304 Physical Applications [15-8 

More precisely the uncorrected observed determinations of m p and m e 
should be the roots of 

10m 2 - 136wm + Hw 2 = 0. (15-84) 

Their ratio is 1834-1. 

We cannot assume that the factor ff will occur in, or have the same 
incidence in, all methods of determining m p /m e or e/m e . The theory of each 
method should be examined in detail in the light of what we have learned. 
In the next section, we shall express our result in a different form, which 
shows the range of its observational consequences more clearly. 

15-9. Revision of the Constants e, m^^m^h. 

In saying that 1847-6 is the correct value of the mass-ratio, and that the 
"observational value" 1834-1 is in error through neglect of a factor in the 
reductions, we are employing the definition of mass or energy which has 
been regarded as fundamental in wave mechanics, namely as the value of 
the operator ( - ih/27r) 3/3J, h being a universal constant. Naturally we do 
not guarantee that 1847-6 will be the correct value if mass is defined in some 
other legitimate, and perhaps preferable, way. It appears that the deter- 
mination of e\m e by the deflection method leads to the mass-ratio 1834-1. 
It cannot well be supposed that the factor j|f is concerned in this case. But 
the deflection method determines a mass which satisfies the classical 
definition. It is therefore not necessarily in conflict with our determination 
of the mass-ratio according to the quantum definition; though it has an 
important bearing on the larger question whether for general purposes the 
quantum definition of mass is the best to adopt. 

The discovery of the factor J|f creates a new situation; and there can, 
I think, be no doubt that the most satisfactory way of restoring order is to 
admit two constants A, h' to be used in connection with internal and external 
wave functions respectively. The remaining constants e, m e , m p have unique 
values. This is a radical change. Prom our first introduction to quantum 
theory we have been taught to regard E = hv as its most inviolable principle. 
But we did, in fact, tacitly abandon it, when we were forced to recognise 
that the momentum operator depends on the index of the wave function, 
and that the index of Dirac's Lorentz-invariant wave function is not the 
same as that of the commonly used wave function of the hydrogen atom. 
Moreover, if any definition has to be altered, h is the obvious victim ; because, 
unlike e, m p ,m e , it does not occur in classical theory. 

We take A' 



Then, if the double-valued internal wave function with operator ( t H/TT) 3/3a? 
is taken to be of index 1, the double-valued external wave function with 
operator ( - ih'lir) 3/3^ is of index j| . 



15-9] Electric Charge 305 

The two fine-structurfe constants axe 

a=:*c/27re a = 137, a'^A'c/ 2 ^ 2 ** 136. 

The fine-structure constant is merely a name for the number of dimensions 
of the double phase space concerned. 

As compared with our previous results, external masses will be reduced in 
the ratio jff, and internal masses will be unchanged. Counting ra as an 
external mass, our previous results become 



Hence m p , m e satisfy ( 15-84), and the mass-ratio is 1834- 1 . Bond's correction 
has been eliminated by the change in the definition of mass. It was originally 
required to reduce internal energies to the unit which was used for external 
energies; but we have now redefined external energies in a way equivalent to 
changing the unit. 

Let us consider the deflection method of determining mass. It would 
ideally be possible to determine the ratio m p jm e by comparing the deflections 
of a proton and electron projected with known energy in the same magnetic 
field. The motion will be in accordance with the wave equation 

(S# A -m)0 = 0, with p IA = (~ih'l27T)dldx (A + K lA , 
the wave functions being external. As h' will not appear in the ratio of the 
two deflections, the fact that it differs from h will not be revealed. The ordinary 
calculation will therefore give the correct mass-ratio, which on the new 
system is 1834-1. 

So far as radiation is concerned, the normal constant h is applicable; for, 
in absorption and emission, radiation is connected with the quantised, i.e. 
internal, states of material systems. 

Thus in general the unconnected observational determinations of the 
natural constants should be consistent with a = 137, m p /m e 1834-1. Either 
they depend only on classical theory, and do not introduce A; or they depend 
on internal wave functions or radiation frequencies. To introduce h' we 
require an experiment determining the absolute wave length of the external 
wave functions of electrons or protons. Presumably the diffraction of 
electrons by matter involves A'; arid the ordinary calculation will give the 
scale of the diffraction pattern ff-J times too large, unless the effect is 
concealed by a compensating factor. At present it is not possible to attain 
this accuracy. It is difficult to devise any other experiment in which the 
factor could manifest itself. 

Summarising our conclusions we find that the definition of energy in 
wave mechanics by the formula E=hvis not in all cases consistent with the 
established meaning of the term "energy" in classical theory. In this section 
we restore the classical reckoning with e, m p ,m e a& fundamental constants. 
We calculate the ratio m p /m e to be 1834-1. The ratio e/m p or e/m e must be 



306 Physical Applications [15-9 

found experimentally since it involves our arbitral^ standards of mass and 
length. It can be found from the Faraday constant, by the oil-drop method, 
or by deflection methods, none of which involve h. We must further consider 
how results that have been expressed in terms of h are to be reduced to 
determinations of e, m p9 m e . We first, by using a=Ac/27re 2 , express them in 
terms of a instead of h. Then a denotes the number of dimensions of the 
double phase space of the wave function concerned and its comparison fluid, 
and may be either 137 or 136. In the experiments which furnish the most 
accurate values of the constants the value is 137; but the theory of each 
experiment should be scrutinised with this point in view. 

This summary covers most experimental determinations of the con- 
stants; but there is one important experiment "which introduces new con- 
siderations. We shall treat it in the next section. 

15*95. The Crystal Grating. 

It is now considered by leading authorities that the various observational 
methods of determining e, m p9 m e , h cannot be completely reconciled by 
current theory. The discrepancy, as usually stated, is that the value of e 
found by the crystal grating method is definitely greater than that found by 
other methods. To put the blame on e is merely conventional; the dis- 
crepancy is in the relations of the constants, and can equally well be attri- 
buted to m p . The question naturally arises whether this discrepancy has 
anything to do with the factor jff. 

We must first exhibit the disagreement in a form free from irrelevancies. 
The crystal experiment consists in comparing the diffraction of the same 
X-rays by (a) a ruled diffraction grating, and (b) a natural diffraction grating 
formed by the lattice structure of a crystal. The comparison determines the 
lattice interval in terms of the known interval between the lines of the 
ruled grating. Thus we obtain the dimensions of the lattice in centimetres, 
and hence the number of lattice cells in unit volume, or in unit mass of the 
crystal. Since the lattice is formed by molecules, this gives the number of 
molecules in unit mass of the crystal. Using the known relative atomic 
weights, we deduce the mass M of a hydrogen atom in grams. 

This appears to be the only practical way of determining M without 
introducing e or h. But we can make another determination via e. A theo- 
retically simple way of finding e is to measure macroscopically the charge 
2ne provided by a large number of a particles; the number n of particles, 
whose charges are collected, is ascertained by actually counting them. 
Further, e/M can be found by an electrolytic method equivalent to accumu- 
lating the charges of a known mass of hydrogen until an amount large enough 
to measure macroscopically is obtained. By combining e and e/M, a value of 
M is obtained which is less than that found by the crystal grating method. 
The observed ratio agrees with f|f within the limits of observational error. 



15.95] Electric Charge 307 

There appears to be gnly one vulnerable link in this chain of connection, 
namely the assumption that in the crystal there is one molecule to a lattice 
cell. That depends on a classical picture of the particles in the crystal. In 
wave mechanics the structure of the crystal is represented by standing 
waves. It is not necessary to suppose that the "particles of composite 
identity", which are more or less located at the lattice points, represent an 
equal number of pure particles. In more elementary problems the invariance 
of the density of eigenfunctions secures that different modes of analysis into 
particles yield the same total number of particles; but this does not apply 
when the number of dimensions of the phase space is different in the two 
analyses. A crystal is a typical example of a system composed of indis- 
tinguishable elements. Any pair of elements will have a permutation co- 
ordinate and an interchange energy conjugate to it. It is possible, and even 
probable, that in constituting the standing waves of a system of this 
character, 137 particles of pure identity furnish 136 composite particles. 

The problem lies rather beyond the point which our theory has reached. 
But if the observations are trustworthy, it would seem to be a direct deduc- 
tion from them that there is a discrepancy in the two ways of counting 
particles in a macroscopic aggregation. In counting a particles, no question 
of interchange arises, because they are observed one at a time. Ideally we 
might vaporise the crystal, and count its molecules in the same way; but 
instead, we try to count the molecules as they lie simultaneously in space. 
It is scarcely surprising that we should become confused in our count of 
entities which have no definite position and are indistinguishable from one 
another. We count something; but not the particles which we should count 
if they passed successively before us. In the crystal we identify a molecule 
by the lattice cell which it occupies, disregarding the fact that the molecules 
counted by the other method could not be localised individually. This 
procedure introduces a permutation coordinate. The extra coordinate 
increases the energy invariant of a particle in the ratio yf |, in accordance 
with the law of equipartition of energy. We have thus to assign to each crystal 
molecule a mass ^ff times that of a corresponding gas molecule as the 
observations actually indicate. This is not energy of crystallisation (which 
is too small to be considered); it is presumably a sign that, in the mathe- 
matical transformation of the external wave functions of gas molecules 
into internal wave functions of a crystal, 137 gas particles become re- 
partitioned in 136 crystal particles. We can thus suggest a possible origin 
of the discrepancy of the crystal results ; but it would require a much fuller 
investigation to determine whether the factor jf^ is an observable correction, 
or whether it is compensated in the complete theory of the experiment.! 

f I understand that (in view of still more recent experiments) the present opinion is 
that the supposed crystal grating discrepancy is spurious. 



CHAPTER XVI 
THE EXCLUSION PRINCIPLE 

16-1. The Second Quantisation. 

Let J r (r=l, 2, ... n) be n commuting idempotent symbols. Let U be a 
common eigensymbol of the J r , and let e r be the corresponding eigenvalue 
ofJ r . Since J r a -J r = 0, e r = or 1. 

By 3-7 (e) an eigensymbol for any given set of eigenvalues e r can be 

found explicitly, namely 

(16-11) 



where V is any symbol. Since J 8 2 J 8 : =0 9 and e 6 a e 8 = 0, 

(Js-es)(Js-l + **) = <>> (16-12) 

from which it follows at once that ( J 8 e 8 ) U = for all values of s. 

Let g=n(J r -l + e r ). (16-131) 

i 

We shall regard q as a transformation operator. It transforms any expres- 
sion V 89 which is not in general an eigensymbol of the J r , into 

U 8 = qV 89 (16-132) 

which is an eigensymbol giving the set of eigenvalues defined by q. By 
(16-12), q is singular; there is therefore no inverse transformation. Also 

b y ( 16 ' 12 ) qJ. = qe t . (16-133) 

A case of special importance is 

V^J 8 V 89 (16-141) 

which gives on transformation 



Ze 8 U 8 = X e U 8 , (16-142) 

i i 

where 2 e denotes summation over those suffixes for which e,= 1. 

Similarly, if F = SJ r J 8 K %8 , (16-143) 

we have U = Xe r e 8 U, t8 = X e U rt8 , (16-144) 

the summation S e being restricted to pairs of suffixes both of which corre- 
spond to eigenvalue 1. 

To apply this in physics, we take J r to be an existence operator for an entity 
or condition c r . IfJ r has the eigenvalue 1, r is said to exist in the system 
described by the eigensymbol V\ if the eigenvalue is 0, e r does not exist in U. 
In the system described by V which is not an eigensymbol of J f , the entity 



16-1] The Exclusion Principle 309 

e r is not definitely pre^nt or absent, and e r is said to have partial existence 
in V. Two kinds of partial existence are commonly recognised; r may have 
a probability of existing in F; or, without being wholly in F, it may have a 
component in F. In the former case, what may be called the "degree of 
existence" is expressed by a numerical relation to complete existence; in 
the latter it is expressed by a directional relation. Both are combined in a 
vector which possesses magnitude and direction; and, as we shall see later, 
the general existence symbol J g , which replaces the classical existence 
symbols (1 for existence, and for non-existence), can be represented as a 
strain vector. 

To understand the importance of this investigation, we must recall that 
in the primary developmqpt of quantum theory a system is looked upon as 
the product of its parts rather than as the sum of its parts. We have treated 
double systems in this way in Chapter xn, and quadruple systems (a proton 
and electron with their respective comparison particles) in Chapter xv. 
But for large assemblages the product method becomes unsuitable. If the 
probability of one member of the assemblage vanishes, the probability of 
the whole assemblage vanishes. That takes us far from the macroscopic 
outlook in which the existence of one particle more or less scarcely matters. 
Now it is important to develop a microscopic theory which shall converge 
to the macroscopic theory when the number of particles becomes very great. 
The basis of this new development must be the assignment of additive 
instead of multiplicative characteristics to the parts into which the system 
is analysed. 

To a certain extent we have anticipated this new development, especially 
in Chapters xm and xiv. But we there freely employed the exclusion prin- 
ciple, and we now return to the beginning of the theory to discover its 
foundation. 

In (16-142) we have practically the classical conception of an additive 
characteristic. The quantity U is made up of contributions U 8 from the 
various particles fi , the summation being of course limited to the particles 
which exist in the assemblage to which U refers. The assemblage is,however, 
viewed from the standpoint of selection rather than creation] so that S C C^ 
appears as part of a universal sum ZU 8 . This is now treated as a particular 
case of an assemblage of partially existing particles with characteristics V 8 , 
which are additive in a generalised sense F = J s V 8 , the symbolic coefficient 
J 8 being a measure of the degree to which the sth particle is present in the 
assemblage in the sense explained above. This generalised addition is of the 
same type as that used in expressing a vector as the sum of its components, 



In calling the entities f "particles'* we are redefining the term partide, 
so as to correspond to the conception of a system as the sum of its particles 



310 Physical Applications [16-1 

rather than as the product of its particles. As wet have pointed out, the 
particles in macroscopic systems in equilibrium have composite identity as 
compared with the pure elementary particles originally introduced. 

We shall refer to the present development as secondary quantum theory, 
the multiplicative combination of particles being primary quantum theory. 

16-2, Jordan - Wigner Wave Functions . 

We shall now express V = S J 8 V 8 in a form, due to Jordan and Wigner, which 
is well known in quantum theory. This transformation is not of any particular 
importance from our point of view, %J 8 V 8 being not only the most direct but 
also the most convenient expression of the additive combination rule; but 
it is desirable to show how it is connected with tjie customary outlook. 

In the application contemplated, the suffixes s refer to the pure elementary 
states that can be occupied by particles, and V 8 is the stream vector of the 
sth state. In general a state is only partially occupied; equivalently we can 
say that a particle whose stream vector is V 8 is partially present in the 
system considered. 

Since the states are pure, V 8 is the product of two wave vectors. The 
Jordan-Wigner theory is limited to the case in which the stream vector of 
each state is a perfect square. We can therefore take V 8 to be a strain vector 
of index 2 equal to if> 8 . 8 , or a strain vector of index equal to $ 8 . t/t 8 , where 
$ s is the complex conjugate of iff ( 8*6). It is understood that these are outer 
products; it would be inconvenient in the present investigation to indicate 
this explicitly by the asterisk notation. 

The characteristic equation J 8 2 7 8 = of an idempotent operator is of 
the second degree, so that it can be represented by a twofold matrix. We 
shall therefore represent J 8 by the idempotent twofold matrix 



where & is a Pauli matrix, i.e. any degenerate twofold matrix whose 
square is 1. In treating E J 8t \V 8 , we wish to resolve J 8t \ into two factors to 
be associated respectively with the two factors of V 8 . It is possible to express 
J 8t ^ as the product of two 2-vectors; but in the Jordan-Wigner theory a 
different kind of factorisation is employed. Let ^, fl , be three perpen- 
dicular Pauli matrices. Then by (3-86) 



(16-212) 

We can therefore write 

J*X = *A> 1-4,A=A, (16-22) 

where a* 88 &(/* + *) = * (/*-**) (16-23) 

and i 8 is a square root of 1, commuting with the 's. 



16-2] The Exclusion Principle 311 

Since the existence operators J r , J 8 commute, the Pauli matrices used for 
their representation commute; that is to say, we must employ for the 
different particles Pauli matrices ^ , J^, vr and A , 8 , ^, v > 8 , belonging to 
different commuting symbolic frames. 

The Jordan-Wigner wave functions are 



. (16-24) 

Forming their outerf products TT and TT, we obtain, if we omit the 



, 

=A,J 



V ' 



by (16-22). We call A the complementary stream vector to V. It represents the 
stream vector of the states to the extent to which they are unoccupied, in 
the same way that V is the stream vector of the states to the extent to which 
they are occupied. 

The omission of the cross-terms is to some extent justified when the 
elementary wave functions $ 8 form an orthogonal set, so that their products 
vanish on integration over the whole domain. It is in this integrated sense 
that *PY and V are equivalent. Physically this correspondence is sufficient, 
because we have no right to treat a wave function except as a whole. But, if 
we consider the universe of completely occupied states, given by S V 8 = V -f A, 
we come back to the classical conception of addition ofV 8 . In order that our 
formulae may converge to those of classical theory, it is necessary that 
TT + YT should converge to F-f A locally as well as on integration; that 
is to say, the cross-terms in TT + TT must vanish. 

The point is that when a state extending over a region of space is partially 
filled, the location of the entity which is (partially) occupying it is a matter 
of probability; and we have the complication usual in statistical theory that 
the expectation value of a product is not the same as the product of the 
expectation values. But when the state is fully occupied, it represents a 
definite distribution which can be treated as a classical fluid having a deter- 
minate density at every point. 

The rs cross-term in W + TT is 



Since the 0's commute, this will vanish if 

a r a 8 + a 8 a r =0, a 8 a r +a r a 8 =Q. (16-261) 

Since ^ r and ^ commute, (16-261) will be satisfied if 

V8 + Vr=<>. (16-262) 

f Outer products so far as 0, and $ t are 'concerned. The symbolic coefficients follow 
their own commutation rules found below; ift a and ift a commute. 



312 Physical Applications [16-2 

That is to say, the factors i s in the different termr must be taken to be 
anticommuting square roots of 1. Hence we have also 

a r a, + a 8 a r =0, a r a^+a a a r = (r^s). (16-271) 

Also by (16-22) a 8 a 8 + a a a s = 1. Combining this with (16-261), we obtain 

a r a 8 + a a a r = S r *. (16-272) 

The equations (16-271) and (16-272) satisfied by the coefficients of the wave 
functions (16-24) are known as the Jordan-Wigner commutation rules. 

16-3. Einstein -Bose Particles. 

In the foregoing analysis we have supposed that the states or particles are 
fully distinguished by the additive characteristic^ which is being studied. 
We shall now consider states with a distinguishing characteristic W 8t , such 
that the states W 8l , W 82 , . . . W^ all have the same V 8 . Let J 8t be the existence 
operator for the particle occupying the state HJ,; then (16- 141) is replaced by 

F = S M J 8 ^==SA^, (16-311) 

where # 8 =J sl + J, 2 + ...+ J 8W . (16-312) 

Since the J ri commute and have eigenvalues 0, 1, the eigenvalues of K s 
will be 0, 1, 2, ... m. Its minimum equation is therefore 

A'(A-l)(A:-2)...(A r -w) = 0. (16-32) 

Let A:' = #-im. 

Then, if m is even, (16-32) gives 



ii / 

K' n 

-*m\ 



l + ^Lj = 0, (16-33) 



where r takes successive integral values from - \m to \m. As w-oo, this 
tends to the limit 8inw /r = 0. (16-34) 

Similarly, if m is odd, the limit is cos nK ' = 0. 

Particles, of which there may be any number in the same recognised state 
V 8 , are called Einstein-Bose particles, as distinguished from excluding or 
Fermi-Dirac particles. The distinction is practical rather than fundamental; 
the Einstein-Bose case arises when the recognised states V 8 comprise a great 
number of ultimate elementary states W 8f9 i.e. when the practical classi- 
fication of states is less minute than the theoretical classification. We have 
then to associate a cardinal operator K 8 instead of an existence operator J 8 
with the recognised state, and its eigenvalue gives the number of particles 
in that state. 

The use of K ' instead of K depends on the theory of the self-consistent 
field. We cannot make a large change in the number of particles in a state 
without upsetting the conditions which governed the original analysis into 



The Exclusion Principle 313 

states. Strictly speaking we must know what states are occupied in order 
to determine the controlling field, and a knowledge of the controlling field 
must precede the analysis into states. In practice this means that we 
employ an approximate or average distribution of the particles to calculate 
the self-consistent field. Then we must provide symmetrically for the 
addition of extra particles, and the subtraction of particles allowed for but 
not present. The characteristic to be considered is therefore V'=*ZK 8 V 8 , 
whe^e 2K' is an operator which takes the eigenvalues 1, 2, 3, ... according to 
the number of additional particles definitely present, and 1, 2, 3, ... 
according to the defect in the number of particles. This is fulfilled by 
operators satisfying the characteristic equation shirr/?' = 0. 

The substitution of K a ' for K 8 is equivalent to the substitution J ri \ for 
J 8t in (16-311). We may regard half of the energy (or other mechanical 
characteristic) of the particle as allotted to the self-consistent field and 
thereby contributing to the energies of the particles in other states. Then if 
the particle is fully present we have to add its half of the mutual energy; if 
it is absent we have to subtract the energy wrongly included in the self- 
consistent field. This is provided for by the eigenvalues , J of J J. 

Another point of view is that (/ ) and K' are relative existence 
operators, as distinguished from the absolute existence operators </, K. 
It would be absurd to schedule everything which does not exist. To call 
attention to the non-existence of an entity implies that we had some 
a priori expectation that it would exist. To regularise this outlook we 
contemplate a comparison system consisting of entities e r which have an 
equal probability of existing or not, so that their respective existence 
operators have expectation values J . The object-system is then specified 
by relative operators J r %, which express deviation from the comparison 
distribution. This procedure in secondary quantum theory corresponds 
to that which we have already introduced in primary quantum theory in 
11-6. In connecting the results of primary and secondary theory it is to 
be noticed that the ideal comparison fluid of secondary theory consists of 
a set of half-occupied states, whereas the comparison fluid of primary 
theory is the impermeable background of fully-occupied states formed by 
the unspecified particles of the universe. 

In the usual applications the number m is treated as infinite, so that 
(16-34) applies. A solution is 

K 8 '=-id/d0 89 (16-35) 

where 9 8 is a periodic argument of all operands contemplated. For then 



Thus &>*&*= e- iTrK '\ so that sin7r^' = 0. We thus obtain a representation 

of v ' ** 



314 Physical Applications [16-3 

It is appropriate that a new coordinate 8 should Ve introduced, since the 
existence of states W^ not discriminated by the Values of V 8 implies that there 
exist other characteristics, e.g. momenta 3/30,, unrecognised. A connection 
with classical theory is obtained by setting 



where t is a common argument for all the functions V 8 . Then v 8 is the fre- 
quency of V 8 \ but V 8 is not necessarily a simple harmonic function. By 

(16-361) i 137 

P_ * jj*a. (16.362) 

27T V 8 Ot 



In particular, if V 8 is a momentum vector id/dx^ , the resultant momentum 
vector is 

r 



which may be compared with the momentum T 4 given by the energy 
operator (13-12). 

In current quantum theory Einstein-Bose particles are represented by 
wave functions analogous to the Jordan-Wigner wave functions of Fermi- 
Dirac particles. It is assumed as before that the stream vector V 8 is a perfect 
square. V is written in the modified form 



Since K 8 now operates on one factor only, we must take K 8 = - 2id/d6 8 . 
A possible factorisation of K 8 is 

g t = e** f a 8 = dlda 89 (16-371) 

since a 8 a 8 = 3/3 (log a 8 ) = - 2i d/d0 8 . Thus, if cross-terms are omitted, 

F' = W = (Sfe)(Za^). (16-372) 

Further, if/ is any symbol, 



so that a 8 a 8 a 8 a 8 =l. And since all other combinations of a 8 , a 8 commute, 
the complete commutation rules are 

(16-381) 
0. (1,6-382) 

These may be compared with the Jordan-Wigner commutation rules in 
(16-271), (16-272). 

In problems concerning a specified system, the unspecified particles will 
in general be Einstein-Bose particles. For it is clear that the type of analysis 
into states V 8 applied to the specified system does not provide anything like 
enough states to accommodate the 10 79 unspecified particles separately. 



16-4] The Exclusion Principle 315 

For example, consideibthe box-problem ( 13*2) applied to material of the 
density of water. If (to avoid introducing supernatural barriers) we suppose 
the uniform density to continue indefinitely, we obtain a space whose radius 
is of the order 3 . 10 8 km., and a mass about 10~~ u of the mass of the universe. 
In this case the recognised states provide room for only 1 in 10 14 particles; 
so that in (16-312) m=10 14 . The unspecified particles provide the imper- 
meable background with occasional vacancies which we have identified 
with photons. Any number of vacancies up to 10 14 may occur in one recog- 
nisedenergy state; and the photons accordingly obey Einstein-Bose statistics. 
It will be seen that a change of outlook can have the effect of changing 
Fermi-Dirac particles into Einstein-Bose particles. The theory of Chapter 
xivis based on a system of excluding particles (system A); but these become 
transformed into Einstein-Bose particles in the usual outlook (system B). 

16*4. Relation to the Energy Tensor. 

It is important to realise the considerable change of outlook that is involved 
in passing from the primary to the secondary quantum theory. The analysis 
which shows that in given field conditions there will be a series of steady 
states, composed of probability distributions with wave functions tf/ a or 
stream vector functions V 8 , is common to both. In primary quantum theory 
a particle may have its total probability (unity) distributed between different 
states, so that its stream vector is V = Zp 8 V 8 . If a second particle is present 
with the stream vector V = p a ' V 8 , there is no warrant for forming the sum 
(p^ +PS) V a . The only combination of V and V contemplated is the product 
FF', which indicates a set of double states V r V 8 ' occupied with probabilities 
PrPa- If *b e addition p 8 +p a ' were admitted, the total probability of a state 
might exceed unity, and the exclusion principle would be violated; this 
contingency does not arise, since the conception of such an addition is 
foreign to the outlook of primary quantum theory. In secondary quantum 
theory a different type of particle is introduced (composite from the point of 
view of primary theory), each particle being limited to one state. The 
numerical probability factor p a is now replaced by a vector probability 
factor </ 8 . The exclusion principle is represented by the fact that, the eigen- 
values of J 8 being and 1, its expectation value will always lie between these 
limits. 

The general theory of relativity throws light on the meaning of V = J 8 V 8 . 
Macroscopically the additive characteristic of a system is its energy tensor. 
Thus the secondary quantum theory rightly rejects the sum of vectors 
f or Zp 8 V 8 , and employs the sum of tensors of the second rank %J 8 V 8 . 
Moreover, owing to the non-linearity of the equations, the self-energy tensors 
of particles are not strictly additive in general relativity theory. The additive 
energy is a mutual energy of the particle and the rest of the universe. The 



316 Physical Applications [16-4 

energy is therefore not the square of the momentum vector V 8 V 8 , but the 
product of a special factor V 8 and a world factor J 8 . We have already seen 
how we endeavour to reconcile this with our usual non-relativistic outlook 
by artificially breaking up the mutual energy into self energies of the 
particle and the rest of the universe (treated as comparison fluid). The 
convergence of the energy tensor %J 8 V 8 of secondary quantum theory to 
the energy tensor T^ v of general relativity theory, as we pass from the 
microscopic to the macroscopic outlook, is exhibited in (16-363). Since 
the macroscopic outlook implies a much less minute discrimination of 
individual states, the quantum expression for the energy tensor is first 
reduced to Einstein-Bose form S K 8 V 8 . 

t 
16*5. Enumeration of Wave Functions . 

By using the observed values of e/m and /c, we have found that there are 
approximately 136.2 256 protons and an equal number of electrons in the 
universe. This will now be confirmed by an independent investigation which 
shows that the number is exact. 

Our ability to predict the number of protons and electrons in the universe 
implies that the number is imposed by the procedure followed in analysing 
the interrelatediiess of our experience into a manifestation of an assemblage 
of particles or wave systems. It is a commonplace that electrons are not 
intrinsically distinguishable from one another; it is therefore not surprising 
that the total number, allowed for in our scheme of dissection of phenomena, 
depends on the conventional distinctions introduced when, for example, we 
decide that a certain diffuse wave packet is composed of two electrons 
rather than one. 

We first notice that the value we have found for the total number of 
particles N belongs to a series of numbers 

^=2.3.24, #2 = 2. 10. 2 16 , JV r 4 =2.136.2 256 , 

i.e. n (n + 1 ) 2* 2 , with n = 2, 4, 1 6, .... If ^/i is regarded as characteristic of a 
simple wave function, N^ and N^ will be the corresponding characteristics 
of double and quadruple wave functions. We shall first investigate a "uni- 
verse" described by simple wave functions and show that it consists of N 
(i.e. 96) particles. It will then be comparatively easy to show that the actual 
universe contains N 4 particles. 

Conditions we can scarcely call them phenomena which require for 
their representation no more than a simple matrix frame have been treated 
in Chapters i to vin. We can represent a position vector or a stream vector, 
but not both; if the position vector is represented in an ^-frame, we have to 
change to a (/-frame to represent the stream vector ( 10-5). The familiar 
particle, whose position vector and stream vector are both inexact but not 



16-5] The Exclusion Principle 317 

entirely uncertain, cannot be represented on this plan. But in Chapter vm 
we made a new departure by introducing D-operators by means of which 
we were able to define vector functions instead of isolated vectors; and in 
particular the momentum vector was defined. In fundamental investigations 
of the kind which we are now treating, the stream vector and the momentum 
vector coalescejf but their coalescence is, strictly speaking, 



For the operator id/dx^ applies to a representation of positions, and it is 
necessary to change to another frame to obtain a representation of the 
stream vector j^ . 

Consequently, by using position and momentum instead of position and 
stream, we are able to keep the whole representation of the particle within 
one matrix frame at the cost of using differential operators as well as the 
matrix operators of the frame. 

This is an illustration of the use of differential operators to avoid multi- 
plicity of frames. It can be greatly extended; and, as is well known, particles 
are generally distinguished by allotting to each a different orthogonal wave 
function, rather than by allotting a different matrix frame as contemplated 
in 2-9. It is true that we must still assign to them existence operators J 8 
which are represented by Pauli matrices in different frames; but that, as it 
were, segregates the question of multiplicity of frame from the main part of 
the analysis, and eliminates it altogether in the case of particles having 
complete existence in the system (J 8 = 1). 

In the systems containing many particles which we have hitherto treated 

the box problem in Chapter xin and the Einstein universe in Chapter xiv 

we were concerned only with fully occupied states, so that existence 
operators (other than unity) with their attendant multiplicity of frames did 
not appear. If the scalar state is not fully occupied, we must in general 
express the extent of its occupation by an idempotent operator J. In (13-61) 
we divided the scalar state into two substates J and 1 J, with equal and 
opposite spin. The purpose of dividing an entity into two parts is that we 
may conceive of one part existing without the other; so that the division is 
made by distinguishing a part J 'which is occupied (since J. J= 1 . 7) from 
a part 1 J which is unoccupied (since J ( 1 J) = 0). The further separation 
of the state into substates with opposite charges depends on a double 
existence operator J r J s ' for two entities, since charge is meaningless unless 
there are two charges to interact. The four substates (13-62) are represented 



t We regard non-algebraic wave functions, whose momentum vector has matrix com- 
ponents and is not to be identified with j^, as a side-development of the wave functions 
primarily studied. As explained in 9*5, the steady states defined by them are only con- 
ditionally steady. 



318 Physical Applications 16-5 

The system represented by a simple wave functjpn has four dynamical 
coordinates, and in field-free conditions we may take the steady state to 
be (cf. (8-53)) ^^ ^ ^ ^ e ^*M**i^*^*4fa 9 (16-52) 

where ^ is a common eigensymbol of the four commuting ^-symbols. The 
dynamical angular momenta i3/3a l9 etc., have eigenvalues . Each set 
of four values defines a corresponding eigenfunction 0, and the 16 com- 
binations of sign therefore define 16 eigenstates. 

In our earlier treatment we regarded a particle as distributed with pro- 
bability factors p r between the different eigenstates; to introduce a second 
particle would involve beginning the problem over again with double wave 
vectors. In secondary quantum theory the states are considered to be 
occupied by 16 different partially existing particles, which in special 
cases may be definitely present or absent. We have therefore to assign 
to them existence operators J r (r=l, 2, ... 16) stating the degree of 
occupation. 

The expression (16*52) singles out a particular plane J5 23 as the plane of 
spin, and this choice will be reflected in the value of the eigenvector . The 
choice might equally well have been E 3l or E^ . This will be investigated in 
greater detail in the next section; but it is fairly evident that the effect of 
these alternatives will be to triplicate the set of independent wave functions, 
making the total number 48. This number is again doubled when we allow 
for two possible algebraic frames (1, i), or an equivalent duplexity. The 
total then becomes 96 or JV^. 

We now begin to see the make-up of the numbers N l9 N 2 , N. The 
elementary states correspond to wave functions of the form e** (:Md ^ . 
The factor 2 w2 corresponds to the variety of the exponential; and the factor 
n(n + 1)/2 corresponds to the variety of ^r ; finally there is a factor 2 corre- 
sponding to the duplexity of the algebraic frame. We notice that in N l9 the 
four dynamical coordinates give a factor 2 4 , whereas the three geometrical 
coordinates give a factor 3. This is because the wave function depends on a 
combination of eigenvalues of the dynamical momenta and a choice of 
geometrical axis. Similarly in N 2 and N^ the large factor arises from the 
number of combinations of eigenvalues of operators with a common eigen- 
symbol, and the small factor represents a choice of "orientation" of this 
eigensymbol. 

The final doubling of the number is the most difficult step to investigate. 
It is fairly obvious that the duplexity of the algebraic frame will contribute 
a factor 2; but it is less easy to be sure that the factor has not already been 
included in the enumeration. 

Before studying the method of enumeration more rigorously, we must 
make clear the connection between the existence of N eigenfunctions of 



16-5] The Exclusion Principle 319 

type (16*52) and the formation of a corresponding universe of N particles. 
In Chapter xrv the particles constituting the Einstein universe were 
represented by wave functions in a flat space, formed by stereographic 
projection of the "actual" spherical space. Here we begin at the other 
end of the problem; for the wave functions (16*52) are those occurring 
in elementary quantum theory, which, as we have seen (14*3), pre- 
supposes an Einstein universe as background. These wave functions 
occupy actual space; and the proper energy of the N l particles, being 
represented by the curvature of space, cannot be represented a second 
time in the wave functions themselves. By the dynamical theory these 
eigenfunctions give the only possible steady states of disturbance of the 
system; there are therefore just N independent steady modifications 
(generalised states of rotation). But, owing to the symmetry of the con- 
ditions, there is a degeneracy which permits us to re-analyse their most 
general combination into various alternative sets of N^ elementary states ; 
in particular we can analyse it into an equivalent number of spherical 
harmonic distributions. By projecting these into a flat space, we restore 
to the wave functions the proper energy which was abstracted to provide 
the curvature, and obtain wave functions corresponding to those of 
Chapter xiv. The latter, by definition, correspond to steady states in 
the self-consistent field produced by the whole aggregation, so that they 
must conform to this method of enumeration. 

Lemaitre has pointed out to me that the ambiguity <E 16 <x 4 in (16-52) 
seems to be in conflict with (8*54), where the algebraic dynamical co- 
ordinate was given unique sign. But this ambiguity is seen to be necessary, 
when we remember that the set of wave functions, when fully occupied, 
must not provide a resultant proper energy additional to that represented 
by the curvature of the space to which they are referred. In primary 
quantum theory reversal of the algebraic coordinate would turn the 
particle into a minus-particle; and here the double sign will imply that 
in some sense the 96 composite particles can be regarded as 48 plus- 
particles and 48 minus-particles though we must not too hastily assume 
that the latter are positrons and negatrons. Since we are treating a set 
of particles whose resultant energy tensor provides the curvature of the 
space in which their individual wave functions are represented, the con- 
tributions of these wave functions to the energy tensor must cancel out, 
and there must be as many minus-particles as plus-particles in the set. 
It is therefore right to include the minus-particles in the enumeration. 
This relative representation provides the simplest method of counting the 
independent wave functions; but to obtain the particles ordinarily re- 
cognised in physics we must restore to them the energy abstracted to 
form the curvature. By this transformation they all become plus-particles, 



320 Physical Applications [16-5 

as in 14-2. To put the conclusion in another form -/'-elementary quantum 
theory always describes its systems as additions to the impermeable 
background which constitutes the Einstein universe. If therefore we 
describe by the wave functions of elementary quantum theory the set 
of particles which composes the impermeable background, the plus- 
additions and minus-additions must balance. But the minus aspect is only 
relative, and the particles are all positive contributors to the total energy. 

16*6. Geometrical Representation of J r . 
By squaring (16-52) we obtain the strain vector of index 2 

F(a l9 a a , a a , a 4 ) = e l ' a i^ a ia 3^4^, (16-611) 

the matrices being eliminated because V Q is an eigensymbol. Since ^ = <A ^o> 
it has the pure form F = J (i + j (i + tf w) . (16-612) 

We can distinguish the possible functions V as V r (u,v), the suffix r 
(r= 1, 2, ... 16) indicating a particular combination of signs in (16-611) and 
the arguments u, v indicating planes in the two Pauli frames. 

Consider the existence operator J of V. J is represented in a Pauli frame; 
and since there is a possibility that V may be an eigensymbol of J, i.e. that 
/F = V, this must be one of the two frames of V. Thus the and frames 
have the fundamental distinction that one of them (namely the spin frame ) 
is also the frame of the existence operator. To interpret this physically we 
recall that J is the world factor in the energy tensor JV ( 16*4). In the 
conditions represented by a simple wave tensor the "rest of the universe" 
is understood to form a uniform static distribution defining an electrically 
neutral three-dimensional space. On the other hand V involves four dimen- 
sions and is not electrically neutral; thus it contains an additional factor 
(1 + iO v ) whose plane v is not definable by reference to the three-dimensional 
space of J. 

Let J W =%(1 +i w ) be the existence operator of a particle w . By (16-612) 
J W F = F if =,, and J W F = if = -. What we have hitherto called 
the degree of existence of the particle w in the system F, is reduced to a 
directional relation between the spin planes w , M of e^and F. This simplifica- 
tion is due to the fact that F is a pure strain vector, and could itself be 
regarded as representing an elementary particle u with an existence operator 
J u . In our original formulae ( 16-1) F was not restricted to be a pure strain 
vector; it might therefore have a constituent agreeing with e w in direction 
but reduced in intensity by a numerical probability factor. This mode of 
partial existence of w in F is now excluded by the condition of purity. 
Accordingly the degree of failure of one elementary particle or wave system 
to exist fully in another elementary wave system is represented exclusively 
by angular deviation, and not by fractional probability. 



16-7] The Exclusion Principle 321 

To enumerate the independent eigenfunctions, we must replace V r (u, v), 
which has a continuously varying parameter u, by an appropriate number 
of functions V r (u a , v) with fixed parameters u a ^ The continuity of u is a 
degeneracy due to the symmetrical conditions postulated, which permit 
relativity rotation in three dimensions (cf. 13*6). To obtain a corresponding 
non-degenerate problem, we should limit the spin to three coordinate planes. 
Since the spin can be in either direction in each plane, this gives six alter- 
natives. We conclude therefore that the 16 eigenfunctions V r (u,v) have a 
sixfold degeneracy, and correspond to 96 non-degenerate eigenfunctions. 

The conclusion that the degeneracy is sixfold, not threefold, can be 
reached in another way. Pauli space is not an ordinary space of three 
dimensions. The Pauli matrices ^, f^, f correspond to JE? 23 , E^, E 31 , which 
have a different group relation from the matrices E ,E 29 E B which represent 
rectangular axes in ordinary space. Suppose that rectangular axes Ox, Oy, 
Oz are laid down geometrically in a three-dimensional space. Taking it to 
be an ordinary space, we associate matrices E l , E 2 , E 3 with Ox, Oy, Oz, 
respectively; in so doing we do not give any chiral quality to the space. 
Taking it to be a Pauli space, we associate Pauli matrices A , f^, with 
Ox, Oy, Oz, respectively; these satisfy ^ v = x = ?,* (3-86). But we 
can also associate with the same geometrical axes an alternative (left- 
handed) Pauli space by using the matrices in the order ^, f v , i^. Treating 
tt as a vector referred to rectangular axes, V r (u, v) will have a threefold 
degeneracy; but there will also be a duplicate set of 48 eigenfunctions 
whose existence operators (or spins) are represented in the alternative 
Pauli space. 

It might be suggested that all possible reversals have already been 
provided for by the signs in (16*52). But a consideration of the hydrogen 
atom makes it clear that the reversal of spin is additional. The orbital motion 
of an electron in any plane may be in either direction. In addition, two 
states are discriminated according as the electron spin is with or against 
the orbital motion. From this aspect the factor 2 arises from the duplexity 
indicated by the "fourth quantum number". 

16*7. Double and Quadruple Wave Functions. 

The enumeration of independent double wave functions proceeds on the 
same lines as the enumeration of simple wave functions. The maximum 
number of commuting symbols E^F V is 16, obtained by combining an anti- 
tetrad of E -symbols with an antitetrad of JT-symbols. We have therefore 16 

t The direction of v in its 3-space does not matter. There is no reference standard to 
compare it with; and, whatever the direction may be, it is called the time. (It should be 
borne in mind that the physical ideas illustrated by a universe composed of simple wave 
functions are necessarily somewhat crude.) 

ETP 21 



322 Physical Applications [16-7 

dynamical coordinates a^, and the double strain /ector function for a 
steady state, corresponding to (16-611), is 



where JJ> is a common eigensymbol of the 16 commuting JS^-symbols. The 
sixteenfold ambiguity provides 2 16 combinations. 

First suppose that V is the product of two simple strain vectors. Then in 
place of (16'612) we have 



An existence operator for a system of two particles is of the form 



since the system does not exist unless both the particles exist. The two Pauli 
frames f , ' will combine into a frame of fourfold matrices JSy/, and the 
frames 6, ff into another frame JJ/.f Then K is a simple strain vector in 
the W -frame and V is a double strain vector in the E'F'-fmmQ. 

The strain vector V is by definition a perfect square V F Y , but in general 
Y is not the product of two simple wave vectors. The system represented 
by a double wave tensor is a superposition with different probabilities of a 
number of resolvable double systems. The essential result for our purposes 
is that the geometrical relations, which replace the conception of partial 
existence, are contained entirely in the JP'-frame. Rotation in the JF'-frame 
can be ignored, since there is no reference standard in that frame. 

We have then a series of wave functions lj([im'], [vv']) 9 r= 1, 2, ... 2 16 , 
involving a continuous parameter [uu']; and we have to determine the 
degree of degeneracy implied by the continuous parameter. If V r represented 
two completely separate particles, [uu f ] would represent their planes of spin 
in three dimensions. Since the particles are not separate [uu f ] is a simple 
strain vector in an ordinary fourfold frame. This brings us to familiar 
ground. Although the domain in which we picture this vector is not ordinary 
space-time, the problem is the same mathematically. The relativity rota- 
tions, whose existence brings about the degeneracy, are the 10 rotations in 
five dimensions or the ten displacements in phase space, which we have 
studied fully in the earlier chapters. There is accordingly a tenfold degen- 
eracy; or, including the duplication due to right- and left-handed frames of 
E' each V r is equivalent to 20 non-degenerate wave functions. This gives 
the total # 2 =2. 10. 2 16 . 

It is perhaps not obvious that the number of rotations to be taken into 
account is 10 not 16. We might appeal to the formal argument that the 
corresponding factors in N^ and N 2 should be either 3, 10 or 4, 16; and since 
the factor in JVi is certainly 3, the factor in N 2 must be 10. But the result is 

t This regrouping has a resemblance to the crossing of frames in 10*4; but it is not the 
same transformation. 



16-8] The Exclusion Principle 323 

verified at once, whence notice that in comparing our double system with 
a standard system of two particles defined by the existence operator K, we 
cannot distinguish between the combination J W J W ! and J^J w f > Thus our 
reference system does not distinguish between a rotation t,^ v f and ,,/, 
and the rotations (^ ' ,,/ must be ignored in the same way that we 
ignore rotations in the 00' frame. Adapting the theory of crossed frames in 
10-6 to twofold matrices, the excluded rotations are the antisymuietrical 
(tinje-like) rotations in the crossed frame; there remain the ten symmetrical 
(space-like) rotations. 

The extension to quadruple wave functions does not raise any new point. 
The reference system of four separate particles defined by a quadruple 
existence operator has a l<J6-dimensional phase space, so that the degeneracy 
is 136-fold. 

In each case the degeneracy of the wave function corresponds to the phase 
space of a tensor of half its rank. A corresponding reduction occurs in the 
theory of the steady internal states of the hydrogen atom, which can quite 
well be (and commonly are) treated by Pauli vectors instead of Dirac 
vectors; but any deviation from the steady state involves the use of Dirac 
vectors. 

16*8. Four -point Elements of Structure. 

We can observe a relation between two physical entities. To mwtsure a relation 
we must compare it with another relation of the same kind. Thus a measure 
is a relation between two relations and involves four entities.! Formally a 
measurement always refers to just four relata, viz. the terminals of the 
object relation and the comparison relation. But in practice systems of 
combination of measurements have been elaborated which enable us to 
attribute measure indirectly to more complex networks of relationship. 

The basis of measurement is therefore a four-point element of world 
structure. It is on this principle that I have developed the generalised field 
theory in Mathematical Theory of Relativity. % Ultimately the theory of 
atomicity springs from the same origin. 

In the field theory it was necessary to couple with the four-point relation 
a condition of affine geometry, viz. that for infinitesimal relations, if the 
relation AB is equivalent to the relation CD 9 then the relation AC is equi- 
valent to the relation BD. More light is now thrown on this axiom by the 
transformation theory. I think that in attempting to compare or describe 
the difference between two objects, we are impelled to envisage a chain of 
intermediate objects to see, as it were, one gradually changing into 
another. If such a transformation is too far-fetched to be considered, we 

t In special cases two of the entities may coalesce. 
J Chapter vn, Pt. II, especially 98. 



324 Physical Applications [16-8 

are nonplussed, as when asked (in the familiar riddle^the difference between 
an orange and a grand piano. Whether or not this is true of all relations, we 
do in physics confine attention to relations which are represented by trans- 
formations. The axiom of affine geometry then reduces to the condition that 
infinitesimal transformations commute as far as terms of the first order. 

From this starting point we deduce the existence of a measure of world 
structure at each point, given by an affine curvature tensor *jB ftw , which on 
contraction yields a metrical tensor g^ v and an electromagnetic tensor F^. 
The tensor g^ v provides a Riemannian geometry, and with this we define a 
new indirect measure system. A measure is still a relation between relations; 
but we now distinguish sharply between the object relation and the com- 
parison relation. The latter is standardised; anc^ it is no longer a relation 
between specific entities, but is vaguely "contained" in the Riemannian 
geometry. The measure is transferred verbally (and usually also in con- 
ception) to the object relation, so that it appears to involve only two entities. 
Finally in the non-relativistic mode of thought which dominates current 
quantum theory (whether styling itself relativistic or otherwise) the measure 
becomes transferred to one terminal of the object relation, a geometrical 
concept (e.g. an origin of coordinates) being substituted for the other 
terminal. By this devious route we arrive at entities supposed to be endowed 
with measurable properties, e.g. an electron endowed with charge and mass, 
although it requires four entities to furnish anything measurable. 

In wave mechanics a distinction is drawn between observables and 
unobservables. It will here be clearer to call them measurables and un- 
measurables. Current wave mechanics attributes measurables (momenta, 
coordinates, spins, etc.) to single entities such as an electron. The measur- 
ables are, however, four-point relations, which become attached to the 
electron because the other three relata are standardised. In the method of 
wave mechanics measurables are expectation values or eigenvalues derived 
from wave functions; and in attaching the measurables to the electron we 
attach also the wave functions which contain them. Primitively the wave 
function of an electron is that of a four-point element of structure. 

The wave function is quadruple, since it specifies a quadruple probability 
distribution of four entities. We have found that the number of independent 
quadruple eigenfunctions is N = 2 . 136 . 2 256 . It is well known that (with due 
allowance for degeneracy when it occurs) the number of eigenfunctions 
persists in all transformations. By the changes in plan of measurement, to 
which we have referred, the quadruple wave functions become replaced by 
double and finally by single wave functions. Our position has been that, for 
better or worse, current theory has made this substitution; and since it is 
implied in all standard nomenclature, we must accept it. We have seen that 
certain previously inexplicable features in current physics, such as the mass- 



16-8] The Exclusion Principle 325 

ratio of the proton and electron, are impositions, which we are forced to 
accept in order to validate the substitution. Another such result is that by 
substituting N^ simple wave functions for the N quadruple wave functions, 
2 . 136 . 2 266 simple wave functions are forced into a space-time which is built 
to accommodate 96. The result of this overcrowding is that instead of their 
wave lengths being naturally adjusted to the curvature of space, they occupy 
spherical harmonics of high order and short wave length as explained in 
Chapter xiv. 

Our result, that the "number of particles in the universe" is the number 
of independent non-degenerate eigenfunctions in a quadruple wave system, 
thus comes from the fact that a measurement, or comparison of relation with 
relation, is an expectation value determined by a quadruple wave system. 

The number of dimensions of space-time can be regarded as one of the 
numerical constants of nature. Effectively, the number 4 was assumed at 
the beginning of our theory, when we chose a symbolic frame based on 4 
anticommuting square roots of 1. But we can now determine it in the 
same way as the other natural constants, and show a fundamental reason 
why space-time has 4 dimensions and no more. 

Space-time may be defined as the continuum in which we represent a 
certain type of physical relation called "displacement". The relation 
connects two relata which are conceived geometrically as points "having 
no parts and no magnitude". We need not add that they have position. 
Their relative position is expressed by the relation of displacement already 
mentioned; absolute position is denied. To pass from this geometrical abstrac- 
tion to a physical universe, we must turn the points into physical entities; 
so that "displacement" is now a relation between entities occupying the 
points. These entities are represented by simple (commuting) existence 
operators, which indicate whether in a given operand (representing a state 
of the universe) the points are occupied or not. Nothing more is required; 
for, so far as the relation of displacement is concerned, the only property 
of the entity is that it occupies the terminal of the relation. The simple 
existence operators J, J r give a combined existence operator J J', which will 
also be the existence operator of the displacement; for the relation will not 
exist unless both the relata exist. 

We have now to form a continuum of displacements. This is constructed 
by forming a Group of operators, which includes all operators of the form 
JJ'. It is not necessary that the Group should consist of J J' operators 
alone; but the JJ 9 operators will be distinguished as the pure operators 
of the Group.f It is simplest to consider the matrix representation of the 

t In a Group the transformations consist of the same operators as the field. It would be 
contrary to the ordinary conceptions of transformation theory to limit the transformations 
to pure operators. Consequently the field must be extended to include impure operators. 



326 Physical Applications [16-8 

operators. Since J has two eigenvalues, it can be represented as a twofold 
matrix. Then JJ' is represented by a fourfold matrix, or equivalently by 
an JS?-number. The required Group is therefore that of the ^-numbers. 

This is the simplest solution; but we must also show that more complex 
solutions are excluded. For example, J might be represented by an idem- 
potent fourfold matrix; but such a matrix has been shown to be of the form 
%(l + i). (1 + id), i.e. the product of two idempotent twofold matrices. 
Since J is the product of two simpler existence operators, the entity defined 
by it can be analysed into two parts either of which can exist without the 
other. This case is excluded because it is implied in the conception of dis- 
placement that the entities related by it are like points "having no parts". 
To represent J by a matrix of order higher than the minimum order 2 
implies that the entity defined by it is something more than an "occupied 
point". 

It being established that displacement is represented by an J?-number, 
we can proceed to develop the theory of space as in Chapters TV and vi. This 
gives the result that space-time is a curved four-dimensional continuum, 
and that its signature is 3 + 1. 

16*9. Reaction on Macroscopic Theory. 

For the most part there is no occasion to modify the existing macroscopic 
relativity theory ,f which I believe to be correct so far as it goes (except for 
the important amendment explained in 8*8). The following are the chief 
points that claim attention when it is reviewed in the light of the present 
results : 

(1) There would seem to be a doubt whether Einstein's theory holds 
rigorously for a rotating macroscopic body, e.g. a star. % The whole subject 
is very obscure, but I think the question can be put in the following form. 
Doubtless the particles in a star have attained a statistically steady dis- 
tribution of spin, such that as many spin in one direction as in the opposite. 
But is this balance of spin relative to (a) axes rotating with the material, or 
(b) Galilean axes ? Since the balance is brought about by interactions between 
the particles, either directly or with radiation as intermediary, I should 
conjecture that the answer is (a). If so we have, relative to the Galilean axes, 
a distribution of sources (amounting in the aggregate to a macroscopically 
appreciable source) for which Einstein's theory provides no notation. If the 
answer is (6), the difficulty does not arise. 

t For definiteness, this is understood to be the theory set forth in my Mathematical Theory 
of Relativity, except that important advances have since been made in the cosmical problem. 
Chapter v should therefore be supplemented by B. C. Tolman's Relativity, Thermodynamics 
and Cosmology, pp. 331-488. 

I Doubt of a similar kind has been raised by Sir J. Larmor (Nature, 137, 271 (1036)). 



16-95] The Exclusion Principle 327 

Assuming (a), the C|tse appears to be one in which it is necessary to use the 
Riemann-Christoffel matrix ( 11 -4). The effect, if any, can scarcely be large 
enough to be of macroscopic importance. Its interest lies in its bearing on 
the conservation of energy, since it suggests the existence of a new type of 
field which, if it is not taken into account, would render even the macro- 
scopic conservation of energy imperfect. 

(2) Since g^ v and K O denote measurable properties of macroscopic fields, 
they are expectation values determined by the wave functions of a great 
number of individual particles. We may indicate this average character 



explicitly by the notation g^ v , K a . In general g tLV K ^g VLV K a9 so that in the 
generalised field theory it is represented by a separate symbol ^ Vf<7 .t 
The generalised theory reduces to Weyl's theory if K tlVta ^g VkV K a . Weyl's 
theory is therefore the approximation obtained by neglecting the difference 
between the average of a product and the product of averages. From a 
practical point of view the correction is trivial; but the question of practical 
application scarcely arose in these theories, whose aim was to gain insight 
into the foundations of world-structure. The point brought out in the general 
theory, which is not seen in Weyl's theory, is that the law of gravitation is 
not an additional limitation but is simply the gauging equation; this is one 
of the essential links in connecting the field theory with the microscopic 
theory developed in this book. 

As it happens, we have employed Weyl's theory in a practical way on 
several occasions (12-7, 12-8). To study the electromagnetic field we 
create an artificial field of electromagnetic potential by a gauge transforma- 
tion; just as in Einstein's theory we study the gravitational field by creating 
an artificial field of gravitational potential by a coordinate transformation. 

16-95. Philosophical Outlook. 

We conclude with a brief reference to the philosophical position towards 
which the present results trend. Unless the structure of the nucleus has a 
surprise in store for us, the conclusion seems plain there is nothing in the 
whole system of laws of physics that cannot be deduced unambiguously 
from epistemological considerations. An intelligence, unacquainted with 
our universe, but acquainted with the system of thought by which the 
human mind interprets to itself the content of its sensory experience, should 
be able to attain all the knowledge of physics that we have attained by 
experiment. He would not deduce the particular events and objects of our 
experience, but he would deduce the generalisations we have based on 
them. For example, he would infer the existence and properties of radium, 
but not the dimensions of the earth. 

The mind which tried to apprehend simultaneously the complexity of the 
t Mathematical Theory of Relativity, 93. 



328 Physical Applications [16-95 

universe would be overwhelmed. Experience must be dealt with in bits; 
then a system must be devised for re-connecting the bits; and so on. One 
outcome of this treatment is that the universe is passed through a sieve 
with 3.10 79 holes to render it more comprehensible. In the end what we 
comprehend about the universe is precisely that which we put into the 
universe to make it comprehensible. 

So far as I can trace, the earliest sign of uneasiness among physicists 
about this procedure was shown in connection with the analysis of white 
light by a grating. The analysis of white light by a prism had been looked on 
as a discovery of its composite nature. When the same thing was done with 
a grating, it was seen that we were openly manufacturing the periodicities 
we thought we had discovered in it. As to the composite nature of white 
light, all that can be said is that if a mathematician chooses to analyse a 
function into Fourier components he is at liberty to do so. 

The theory of relativity drew much fuller attention to the subjective 
aspect of many of the laws of physics. The Lorentz transformation comprises 
a number of laws which seem to describe properties of the natural objects 
which surround us (e.g. change of mass with velocity, FitzGerald con- 
traction). But we put these properties into the objects because it is our 
habit to refer everything-to a reference frame in which space and time are 
separated, although there is no such separation to be found in nature. 
Passing to a more advanced illustration of subjective influence, we have 
seen ( 13-4) that the principle of least action arises because (as part of our 
system of comprehending experience in bits) we feel the need to localise the 
various measures which we employ. Localisation is an artificial concept in 
an interrelated universe; and indeed in elementary wave mechanics the 
conception of energy and momentum is primarily introduced as an attribute 
of infinite plane waves, spread over the whole universe. 

In microscopic physics the question of how much we discover and how 
much we manufacture becomes still more acute. We cannot observe a 
microscopic particle without grossly interfering with it. It is often said that 
the particles are put into particular states by the type of experiment we 
perform on them. That is scarcely a fair view of the nature of the inter- 
ference by the observer. Ideally he might wait until the conditions of the 
experiment reproduced themselves naturally. His interference is selective 
rather than active. But so far as physical theory is concerned, it makes 
little difference whether the observer selects the state, or puts the system 
into the state, which he "discovers". Thus before enumerating the cha- 
racteristics of an elementary particle, we have to indicate the type of 
observation by which its existence is supposed to have been recognised. 
Was it the momentum or the position that was noticed? Or was it inferred 
as one of n particles whose total momentum and mean position were found ? 



16-95] The Exclusion Principle 329 

Thus at various stagey in this book we have had to distinguish specified, 
unspecified, and macroscopically specified particles, internal and external 
particles, neutral and vector particles and so on. Their diverse properties 
are the result of the different varieties of observational interference (active 
or selective) which precede our recognition of them. At the beginning we 
treated abstract elementary particles supposed to have been the subject 
of absolute measurements of position and momentum. In a sense these 
particles were all protons or electrons. But only when we recognised that 
actual measurements of momentum and position are relative, did we reach 
particles having the characteristic mass and charge of the proton and 
electron. 

Fifteen years ago I was^responsible for an oft-quoted remark,! tfc It is one 
thing for the human mind to extract from the phenomena of nature the 
laws which it has itself put into them; it may be a far harder thing to extract 
laws over which it has had no control. It is even possible that laws which 
have not their origin in the mind may be irrational, and we can never succeed 
in formulating them." This seems to be coming true, though not in the way 
that then suggested itself. Laws of atomicity have since been discovered, 
and have turned out to be rational and comprehensible to the mind; but it 
turns out also that they have been imposed by the mind in the same way as 
the other rational laws. But a new situation has arisen, because we now 
recognise that the totality of (mind-made) law does not impose determinism. 
Room is left within the scheme of physical law for undetermined behaviour. 
Behaviour whose laws are irrational was perhaps as near to the conception 
of undetermined behaviour as the thought of the time could reach. 

The physicist might be likened to a scientific Procrustes, whose anthro- 
pological studies of the stature of travellers reveal the dimensions of the bed 
in which he has compelled them to sleep. Yet I do not think that we take 
unwarrantable liberties with the universe in our Procrustean treatment of it. 
If experience is a subject-object relation, the subject is entitled to nay, 
he cannot divest himself of his half-share. It can scarcely be a coincidence 
that Heisenberg's uncertainty principle has defined the half-way line with 
mathematical exactitude, distributing a coordinate to one side and a 
momentum to the other side with perfect impartiality. And so we may 
look forward with undiminished enthusiasm to learning in the coming 
years what lies hidden in the atomic nucleus even though we suspect that 
it is hidden there by ourselves. 

t Space Time and Gravitation, p. 200. 



INDEX 



A priori probability, 107, 198 
Absolute and relative displacement, 109, 183 
Action invariant, 136, 243 
Action principle, 238, 240 
Added system, 262, 266, 268 
Addition, non-commutative, 154; general- 
ised, 309 

Additive characteristics, 154, 309, 315 
Affine geometry, 183, 323 
Algebraic coordinate in phase space, 106, 

216, 220 

Algebraic frame, 48 

Algebraic function, 125; wave function, 120 
Algebraic number, 20 
Algebraic wave tensor, 89, 248 
Angular momentum, 122; dynamical, 123; 

quantisation of, 147; macroscopic, 248, 

326. See also Spin 
Antedating, 111, 224 
Anticommuting symbols, 21, 51, 312 
Antiperpendicular rotation, 51, 94 
Antisymmetrical matrices, 40, 96; double 

matrices, 159, 166 

Antisymmetrical tensors, 54, 57, 58, 181 
Antisymmetrical wave functions, 167, 288, 

290 

Antitetrad, 23, 69, 128 
Antithetic, 75 
Antitriad, 23, 69, 128 
Associated strain vector and space vector, 

103 

Asterisk notation, 16 
Augmented phase space, 217, 219 
Average particles, 234 

Basis of statistics, 198, 216, 281, 286 
"Blank sheet", 32, 40, 56, 107, 109, 287 
Bond, W. K, 272, 273, 303 
Born, M., 151 
Born-Infield theory, 136 
Box problem, 235 

Cardinal operator, 312 

CD frame, 161, 167, 203 

Chain multiplication, 16, 155 

Chandrasekhar, 8., 235 

Characteristic equation, 44 

Charge (electric), sign of, 49, 80, 88, 168; 

association with mass, 223; origin of, 283; 

value of e, 304 
Charge-current vector, 90. See Stream 

vector 



Clifford's numbers, 22 

Closure of space, 78, 83, 97 

Cogrodient, 14 

Collision interchange, 245 

Combined systems, 110, 158, 215 

Commutation rules, Jordan- Wignor, 312; 

Einstein-Bose, 314 
Commuting operators (W, (\, (7 a , Z7 3 ), 128, 

141 ; for hydrogen, 144 
Compact .AJ-numbers, 72, 101 
Comparison fluid, 180; partial, 196; neutral, 

206; indistinguishable), 294 
Complementary stream vector, 311 
Complete energy tensor, 163 
Complete momentum vector, 65 
Complete sots, 22, 35, 47, 156; transforma- 
tion of, 27, 160 
Complete space vector, 54; constitution of, 

57 

Complete stream vector, 65, 73 
Complex conjugate wave functions, 131 
Component of an ^-number, 23; formula for, 

37; in double frames, 157 
Composite individuality, 234, 285, 307, 315 
Configuration, 95 
Conjugate triads, 23, 42, 47, 160 
Conservation of probability, 115, 120; of 

energy and momentum, 229, 327 
Constant of gravitation, 273 
Constants of nature, 3; revision of, 304 
Continuous and discrete wave functions, 213, 

226 

Contracting universe, 279 
Contraction of matrix (spur), 36; of double 

matrix, 156; of wave tensor, 206; of 

volume element, 213 
Contragrodiont, 14 
Contravariant, 15 

Cosmical constant (A), 3, 188, 194, 279; in- 
determinacy of, 229, 238, 252 
Cosmical Riemann-Christoffei tensor, 202, 210 
Cosmical system (system .4), 266 
Coulomb energy, 281, 297, 301 
Coupling of spin, 228 
Covariant derivative, 121; with respect to 

X, 289, 300 
Covariant vector, 15; wave tensor, 19, 94, 

109 

Crossed frames, 161 ; significance of, 162 
Crystal grating, 306 
Curvature of space-time, 5, 55, 82, 126, 203, 

256 



332 

Curvature tensor, gauge-invariant, 251 
Cylindrical curvature, 220 



Index 



^-symbols, 125, 317 

Darwin, C. G., 1 

de Sitter space-time, 5, 182, 192, 202, 207, 
211, 264 

Debye-Hiickel effect, 139 

Deflection value of e/m, 304 

Degeneracy formula, 234, 253 

Degeneracy of systems and oigenfunctions, 
5, 247, 292, 321 

Degenerate ^-number, 23 

Description, systems of, 13, 14, 117 

Determinant of an J^-number, 42, 107 

Differential wave equation, 119; strain 
vector form of, 130 

Diffraction of electrons, 305 

Dirac, P. A. M., 1, 6, 15, 36, 62, 90, 115 

Dirac's theory of the positron, 265 

Dirac' s wave equation, 63, 87, 106; dif- 
ferential equation, 119, 130, 144, 151 

Discrete and continuous wave functions, 213, 
226 

Displacement vector, 15 

Displacements, existence operators of, 325 

Divergence operators, 118 

Double frame, 155; wave vector, 155; phase 
space, 166; existence operator, 322, 325 

Double tensors, importance of, 169, 181 ; re- 
duction to simple tensors, 207, 212 

Doublc-valuedness of wave function, 60, 
151 

du Val, P., 36 

Dual of a tensor, 162 

Dual B.C. tensor, 192 

Dynamical coordinates, 116, 123, 127, 140 

Dynamical equations, 126, 128, 144, 213; for 
strain vector, 136 

^-numbers, 21 ; matrix representation (given 

in full), 42 
^-symbols, 22; line as independence of, 24; 

representation by matrices, 36, 39 
Eigenfunctions, enumeration of independent, 

318; double and quadruple, 321 
Eigensymbols and eigenvalues, 44 
Einstein, A., 1 
Einstein-Bose particles, 312 
Einstein space-time, 5, 195, 211, 264 
Einstein universe, 192, 256, 258, 278; 

metastability of, 277 
Electric charge. See Charge 
Electric moment, 89 
Electrical energy tensor, 189, 194 
Electrical matrices, 84; rotations, 84 



Electromagnetic potential (* tt ), 119; gauge 
transformation of, 134; origin of, 137; 
macroscopic character of, 139 

Electron-point, 140 

Electron pressure, minimum, 234, 258 

Elementary particle, 74 

Energy, conservation of, 229, 327; furnished 
by specification, 197; threshold, 262 

Energy invariants, 221, 260, 267 

Energy levels of hydrogen, 147, 303; ne- 
gative levels, 265 

Energy operator (T a3 ), 230, 250 

Energy tensor, 91, 162, 169, 188, 229, 315 

Environment, 32; standard, 129, 143, 150, 
180 

Epistemologv, 5, 328 

Equipartition of energy, 221, 254, 268 

Equivalence of frames, 29, 32; of points in 
space- time, 56, 77; of volumes, 99 

Exact energy tensor, 164 

Exclusion principle, 231, 233, 253; limited 
to steady systems, 246, 254; in secondary 
quantum theory, 315 

Existence, degree of, 309, 320 

Existence operators, 308; double, 317, 322; 
geometrical representation, 320 

Expanding universe, 225, 273, 279 

Expectation value, 38, 149; of spin mo- 
mentum, 122; of energy operator, 231, 
257 

Exponential, non-algebraic, 50 

Extended wave function, 284 

External and internal states, 92 

Factorisation of wave tensors, 66; conditions 
for, 69, 70; in secondary quantum theory, 
310, 314 

Factors, numerical, (J JJ), 10, 302, 304, 307; 
(*), 227, 258, 276; (), 259, 276, 278; 
(2), 143, 152, 261, 274/277, 318, 319 

Factors, symbolic, 37 

Fermi-Dirac statistics, 282; law of anti- 
symmetry, 290; particles, 312 

Field theories. See under Generalised, Uni- 
fied and WeyV* theory 

Final wave vectors, 16, 18 

Fine structure constant, 3, 153, 295, 298 

Five-dimensional relativity, 190 

Five-dimensional space vectors, 63 

Flat space-time, 5 

Four dimensions, space vectors in, 55; reason 
for, 3, 325 

Four-point elements of structure, 170, 323 

Four-point matrices, 34, 38, 96 

Fowler, R. H., 236 

Frame, geometrical, 179 



Index 



333 



Frame, symbolic, 29; right- and left-handed, 
48; neutral and macroscopic, 84; double, 
155; crossed, 161; of identity, 291 

^riedman-Lemaitre theory, 279 

Frobenius's theorem, 46 

Gauge-invariant curvature tensor, 190, 251 
Gauge transformation, 135, 223, 226, 295; 

non-integrable, 137 ; natural gauge, 136 
Generalised field theory, 5, 85, 190, 295, 324, 

327 ; amendment of, 136 
Gravitation, constant of, 3, 273; law of, 

136. 

Gravitational potential energy, 264 
Ground state of the universe, 256, 259, 279 
Group property, 13 

Hamiltonian, 63, 129; reality conditions for, 
87; strain vector form, 106; for internal 
coordinates, 175; for standing waves, 237 

Heisenberg, W., 1. See Uncertainty principle 

Helium, packing ratio in, 10 

Hermitic conditions, 75, 131, 152 

Homothetic, 75 

"Hybrid" particles, 201 

Hydrogen atom, wave equation of, 145; in 
practical units, 153; interaction in, 291; 
evaluation of Coulomb term, 298 

Hyperbolic rotation, 76, 97, 217 

Idempotent symbols, 70; physical import- 
ance of, 91 

Identity, the fundamental, 67; generalisa- 
tion of, 187 

Identity (physical), relativity of, 290; com- 
posite, 234, 285, 307, 315 

Imaginary gauge- transformation, 135 

Imaginary matrices, 40 

Index of wave tensor, 131, 152, 274, 304 

Indistinguishable particles, 282; dynamics 
of, 285; comparison fluid, 293; protons 
indistinguishable from electrons, 287 

Inhibited rotation, 110 

Initial probability distribution, 107, 11)8, 
216, 281, 286 

Initial wave vectors, 16, 18 

Inner product, 15; notation for, 17 

Integrated energy tensor, 253 

Interaction, 281; of system with environ- 
ment, 112, 116, 129; at boundary, 249; in 
hydrogen atom, 291; in general systems, 
154, 298 

Interchange, continuous, 283 

Interchange coordinate. See Permutation 
coordinate 

Interchange energy, 245, 283, 293 



Interchange operator, 158; factors of, 159; 

for double frames, 160; associated with 

Coulomb energy, 301 
Interlocked transformations, 54, 109 
Internal and external states, 92 
Interval, 75; measurement of, 81, 242 
Invariant time (s), 116, 129; represented by 

algebraic coordinate, 106, 216 
Inverted cross, 161, 164 
Irrational laws, 329 
Irregular metric, 211 
Irrevorsibility of time, 117, 225 

Jordan, P., 151 

Jordan- Wigner wave functions, 169, 311 

A' state of a system, 259, 269 
Kinematical energy tensor, 189 
Kinomatical matrices, 84; rotations, 84 
Kinetic energy, distribution law, 246 
Kummer collincation group, 36 

Left-handed frames, 48, 81, 85, 172 

Lemaitre, G., 47, 279, 319 

Light-time, 111, 129, 218 

Limit state, 259 

Linear transformations, 13 

Linkage of system to surroundings, 32, 111, 
117, 129, 292 

Linked rotations, 182, 187; translations, 189; 
displacement in time, 187, 192, 201 

Littlewood, D. K, 71 

Local orthogonal coordinates, 98 

Local system (system /?), 266 

Localisation of characteristics, 240 

Lorentz transformations, 76, 116, 249; in- 
applicable to internal state, 92, 96; to 
Einstein world, 211; to exclusion prin- 
ciple, 231; to non-transferable energy, 
254 

MacDuffee, C. C., 7 

Macroscopic and microscopic theory, con- 
nection of, 4, 169, 201, 208, 256; in secon- 
dary quantum theory, 315 

Macroscopic relativity theory, emendations 
required, 136, 279, 326; new light on, 190, 
327 

Macroscopic set, 83 

Magnetic field, energy due to, 227 ; strong and 
weak fields, 228 

Magnetic moment, 88, 89 

Mass, origin of, 201, 262, 268, 276 

Mass-constant, 230 

Mass-ratio of proton and electron, 3, 219,. 
304 



334 



Index 



Matrices, 16; non-commutative property, 17; 
components, 36; real and imaginary, 40; 
Pauli, 47; kinematical and electrical, 84; 
space-like and time-like, 96; double, 155 

Matrix representation, advantage of, 39, 96 

Matrix representing general ^-number 
(given in full), 42 

Measurables, 170, 324 

Measurement in strong fields, 81, 243 

Motastable states, 150 

Metrical tensor (^), measurement of, 81; 
identified with kinematical self energy 
tensor of comparison fluid, 196, 209 

Microscopic and macroscopic theory, con- 
nection of, 4, 169, 201, 208, 256; in secon- 
dary quantum theory, 315 

Microscopic vorticity, 189, 326 

Milner, S. R., 48 

Minimum equation, 44 

Minor complete set, 47 

Minus particles, 88, 90, 264, 319 

Mixed tensor, 14, 29, 54, 94 

Modifying factor, 107, 281 

Momentum and position, 162. See also 
Uncertainty principle 

Momentum operator, 120, 131; in practical 
units, 152; relation to energy operator, 
230 

Momentum vector, 63; complete, 65; non- 
algebraic, 120 

Monothetic, 75 

Multiplication, inner, outer and chain, 15, 
16; non-commutative, 17; of probabilities, 
154, 213, 309 

Multiplicity of frames, 204, 317 

Mutual energy invariant, 221, 260; pressure 
invariant, 221, 231 

Mutual energy tensor, 165, 205, 315 

Natural coordinates, 98 

Natural gauge, 136 

Nebulae, speed of recession, 279 

Negative and positive charges, 81, 88, 223, 

301; balancing of, 278 
Negative energy levels, 265 
Negations, 90, 265 
Neumann, J. v., 2, 36 
Neutral particle, 89; mass of, 219, 267; 

formula for mass (m ), 271, 275 
Neutral space-time, 80; sets, 83; comparison 

fluid, 206 
Neutron, 7 

Newman, M. H. A., 42 
Nonads, 42 
Non-algebraic phases, 97; wave functions, 

120 



Non-commutaipve multiplication, 17; ex- 
ponentials, &tr, 53; addition, 154; wave 
functions, 311 

Non-integrable gauge transformation, 137 / 
Normalised ^-numbers, 70; strain vectors, 

106 

Nuclei, atomic, 7; disruption by protons, 246 
Number of dimensions of space-time, 3, 325 
Number of particles in the universe, 272, 
316, 325; relation to number of eigenfunc- 
tions, 319 

Numerical constants of nature, 3 

Object system, 180 

Obsorvables, 170, 324 

Observer, 79; his selective interference, 328 

Outer product, 16; notation for, 17 

Packing ratio in helium, 10 

Parallel displacement, 122; generalised, 126; 
in comparison fluid, 182; in \> 288 

Parallel strain, 134 

Parameters of wave functions, 60; promo- 
tion to coordinate rank, 123, 289 

Partial comparison fluid, 196, 198, 207, 210 

Partial existence, 309, 320 

Partially occupied states, 311, 317 

Particles, 8; elementary, 74; neutral, 89; 
scalar, 231 ; number in the universe, 272, 
316, 325 

Particles, external and internal, 174, 221 ; of 
composite identity, 234, 285, 307, 315; in- 
distinguishable, 282, 285; excluding (Formi- 
Dirac) and non-excluding (Einstein-Bose), 
312 

Partition, scheme of, 183 

Pauli matrices, 47, 160, 248, 278, 310 

Pauli space, duplexity of, 321 

Pentad, 23, 35; real and imaginary members, 
40 

Pentadic part, 68 

Permanent, 59 

Permutation coordinate, 153, 228, 283, 284; 
relativity of, 291; recoil, 293 

Perturbation, 116, 141, 239, 249, 292 

Phase, 97 

Phase space, 95; singular, 112; double, 165; 
in CD frame, 166 

Philosophical outlook, 327 

Photons, 263, 315 

Physical reality, invariance of, 76 

Position and momentum, 162. See also 
Uncertainty principle 

Position vector, 73, 162; tensor, 164 

Positive and negative charges, 81, 88, 223, 
301; balancing of, 278 



Index 



335 



Positively saturated space, 81 

Positron, 00, 264 

Potential. See Electromagnetic potential 

I rtential energy, 245, 264 

Pressure in degenerate gas, 235, 236; relation 
to energy-density, 237, 241 ; change of zero 
reckoning, 258, 269 

Pressure invariant, 221, 231 

Probability, 107; distributions of ("states") 
32, 115; conditions of conservation, 119, 
12C; concerned in dynamical equations, 
142; multiplication of, 154,213; of identifi- 
cation, 284, 286; form of partial existence, 
309. See also Initial probability distribution 

Procrustean methods, 329 

Progressive waves, 247 

Projectivo relativity, 56 

Proper time. See Invariant time 

Pseudo-reciprocal, 26, 63, 112 

Pure wave tensors, 65, 68; standard forms, 
69; idempotent condition, 70, 73; physical 
importance of, 91 

Purity lost by contraction, 208 

Quadratic equation for mass, 219, 304 
Quadruple wave functions, 170, 297, 323 
Quantum numbers, ri, 147; u, 147; p, 149 
Quantum theory. See Microscopic theory and 

Wave mechanics 
Quarterspur, 23, 37 
Quotient of ^-numbers, 25 

Radiation, 7; fields of, 143, 262 

Real and imaginary matrices, 40 

Reality conditions, 86; in phase space, 97; 

for gauge transformation, 135 
Recession of the nebulae, 279 
Reciprocal, 26 
.Recoil of comparison fluid, 183; in time, 186, 

191; in *, 293; in r, 296 
Reduced mass, 174 
Reflection of a complete set, 28 
Relata and relations, 170, 181, 323 
Relative and absolute displacement, 109, 

183 

Relative coordinates, 170, 174; energy ten- 
sor, 197, 205; time, 217, 295; identity, 290; 

existence operator, 313 
Relativistic degeneracy, 235, 246, 253 
Relativity rotation (or transformation), 30, 

50, 165, 248; of identity, 283, 285 
Rest energy (or mass), origin of, 262, 268, 276 
Riemann-Christoffel matrix, 187 
Riemann-Christoffel tensor, 181; relation to 

energy tensor, 188; dual of, 192; cosmical, 

202 



Right- and left-handed frames, 48, 81, 85, 

172 
Rotation, 30; antiperpendicular, 51, 94; 

circular and hyperbolic, 76, 97, 217. See 

also Relativity rotation 

Saturated space, 81, 164 

Scalar particles, 231; mass-constant of, 236; 

system of N', 259 

Scalar product, 15; notation for, 17 
Scalar wave functions, 229; relation to 

vector wave functions, 247 
Schrodinger's wave mechanics, 75, 89, 143, 

229, 232 
&P-numbers, 35 
Second quantisation, 308 
Secondary quantum theory, 310, 315 
Self-consistent field, 260, 312 
Self -energy tensor, 162, 165, 205, 209, 315; 

complete, 163 

Solf -pressure and energy invariants, 221, 260 
Signature of space-time, 3, 326 
Simultaneity in internal space, 92, 110; not 

applicable to imaginary time, 295 
Single-valued wave functions, 150, 287 
Singular matrices and symbols, 18, 26; con- 
ditions for, 43, 44 
Singular phase space, 112; transformations, 

173, 308 

Sitter, W. de. See <U Sitter 
Six-dimensional group of rotations, 85 
Six-vector, 57 
Sommerfeld's formula for energy-levels, 147, 

302 

Sound waves, 245 

Space-like matrices, 96; in double frame, 165 
Space tensors, 15; vectors, 53; complete, 64; 

association with strain vectors, 104, 110 
Space-time, four dimensions of, 3, 325; 

goneses of macroscopic, 80 
Specification of particles, 196; energy fur- 
nished by, 197, 267; macroscopic, 201 
Spectral set, 72; analysis of algebraic num- 
. bers, 89, 249 

Spherical curvature (Gaussian), 78, 220 
Spherical space, 5, 55. See de Sitter and 

Einstein space-time 
Spin, 73, 88; momentum, 122; coordinates, 

221, 283; duplexity, 321 
Spur, 36 

Standard environment, 129, 143, 150, 180 
Standing waves, 234, 244; hamiltonian of, 

237 
States, 32, 115; degeneracy of, 124, 247, 289, 

292 
Statistical equilibrium, 141, 239 



336 



Index 



Steady state, 140; generalised, 143 
Stereographic coordinates, 101; application 

of, 218, 219 

Stereographic projection, 257 
Stern-Gerlach effect, 226, 278; in weak fields, 

228 

Straight cross, 161, 164 
Strain vector, 94; association with space 

vector, 104, 110 
Strain vector form of wave equation, 106, 

130 
Stream vector, 65; of elementary particle, 

88; connection with momentum, 122; 

algebraic, 248; in secondary quantum 

theory, 310 

Strong action principles, 242 
Strong and weak magnetic fields, 228 
Suffix coordinates, 283 
Suffixes, omission of, 16 
Summation convention, limited use, 22 
Symbolic calculus, 20, 125 
Symbolic factors, 37 
Symbolic frame, 29 
Systems A, R (cosmical and local), 266; A' 9 

B' t 270 

Temple, G., 27, 73, 145, 151 

Tensor, 13 

Tensor-density, 58 

Tetrode, H., 2 

Threshold energy, 262 

Time, distinctive character of, 75; in in- 
ternal states, 92; irreversibility of, 117, 
225; linkage of displacement in, 187, 192; 
treated differently from spatial displace- 
ment, 201. See also Invariant time 

Time, imaginary relative, 295 

Time-like matrices, 96; in double frame, 165 

Tolman, R. C., 256, 273, 326 

Transformations, etymological, 136 

Transformations, linear, 13; of wave tensors, 
17, 18; of complete orthogonal sets, 27; 
unitary, 43, 98; of strain vectors, 94; 
general symbolic, 125; of gauge, 134; non- 
integrable, 137; of double wave vectors, 
155 ; of double frames, 162 ; associated with 
interchange, 283, 287; of existence condi- 
tions, 308. See also Lorentz transformations 

Transitions, 185, 292 

Translations, 77; linkage of, 189 

Transpose, 19 



Triads, 23; coHugate, 23, 42, 47, 160 
Two particles, representations of, 167 

Uncertain energy tensor, 164, 203, 210 

Uncertainty principle, 74, 162, 169, 179, 191, 

. 275, 286, 329 

Unified theories, 4, 243 

Unit matrix, 18 

Unitary transformation, 43, 98 

Universe, 7; Einstein and de Sitter forms, 

211; analysis of Einstein universe, 256; 

expanding, 273, 279; number of particles 

in, 272, 316, 325 
Unspecified particles, 196 

Vector, 16; basic, 15, 54; space, 53, 54; 

strain, 94; double, 155 
Vector density, 58; three-dimensional, 59, 
105; discriminated by uncertainty prin- 
ciple, 74; by reality conditions, 86, 87 
Vector wave function, 60; relation to scalar 

wave function, 248 

Velocity of indistinguishable particles, 285 
Volume clement, 58, 99; contraction of, 213 
Volume of phase space, 99 ; of spherical space, 
257 

Wave equation, 1, 62, 106; differential equa- 
tion, 119, 130; for hydrogen, 144, 153, 298 
Wave functions, 60, 66; replacement of 
double by single, 212; scalar, 229; w-tuple, 
244; Jordan- Wigner, 310 
Wave mechanics, 1 ; method, not theory, 7, 
66, 256 ; importance of conception of states, 
32, 115; principles of approximation, 112, 
129, 249; a statistical theory, 239; modi- 
fied outlook of secondary theory, 309, 315 
Wave packets, 225, 263, 286; pockets, 263 
Wave tensor calculus, origin of, 2, 15 
Waves equivalent to curvature, 256 
Weak action principles, 242 
Weak and strong magnetic fields, 228 
Weyl's theory, 1, 5, 85, 135, 136, 252, 295, 
327. For the author's generalisation see 
Generalised field theory 
White dwarf stars, 235, 255 

Zanstra, H., 10 
Zariski, O., 36 

Zero reckoning of energy and pressure, 188, 
195, 197, 229, 238, 258, 266, 313 



CAMBRIDGE: PBINTED BY WALTER LEWIS, M.A., AT THE UNIVERSITY PRESS